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- CommentRowNumber1.
- CommentAuthorMike Shulman
- CommentTimeAug 28th 2015
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Author: Mike Shulman
Format: MarkdownItexI don't like the name of the page [[complete topological space]]; it seems to suggest that a property of "completeness" can be defined for topological spaces, when in fact one needs additional structure on a topological space (like a metric, a uniformity, or at least a Cauchy structure) in order to say what "complete" means. Since the notion of [[Cauchy space]] seems to be the maximum generality in which the notion applies, how about renaming the page to "complete Cauchy space"?

I don’t like the name of the page complete topological space; it seems to suggest that a property of “completeness” can be defined for topological spaces, when in fact one needs additional structure on a topological space (like a metric, a uniformity, or at least a Cauchy structure) in order to say what “complete” means. Since the notion of Cauchy space seems to be the maximum generality in which the notion applies, how about renaming the page to “complete Cauchy space”?
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeAug 28th 2015
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Author: Todd_Trimble
Format: MarkdownItexI agree; it's somewhat misleading since completeness is not a topologically invariant property. The renaming suggestion works for me.

I agree; it’s somewhat misleading since completeness is not a topologically invariant property. The renaming suggestion works for me.
- CommentRowNumber3.
- CommentAuthorTobyBartels
- CommentTimeAug 28th 2015
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Author: TobyBartels
Format: MarkdownItexI could have sworn that this page was at [[complete space]] already. That still redirects there. I\'d just as soon use the simpler name. (Of course, the current name is no good.) I\'m not even sure that [[complete topological space]] should redirect to that page. The term ‘complete topological space’, while terrible, is well established for a completely *metrizable* topological space, and there are many results specifically about that topic. So it should probably get its own page, or at the very least be clearly defined on this page.

I could have sworn that this page was at complete space already. That still redirects there. I'd just as soon use the simpler name. (Of course, the current name is no good.)

I'm not even sure that complete topological space should redirect to that page. The term ‘complete topological space’, while terrible, is well established for a completely metrizable topological space, and there are many results specifically about that topic. So it should probably get its own page, or at the very least be clearly defined on this page.
- CommentRowNumber4.
- CommentAuthorTobyBartels
- CommentTimeAug 28th 2015
- (edited Aug 28th 2015)
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Author: TobyBartels
Format: MarkdownItexPage moves as such aren\'t kept in the history, but it seems likely that [Urs did this last June](http://ncatlab.org/nlab/revision/diff/complete+topological+space/8), which is certainly when he added the term ‘topological space’ to the lede (and the redirect from the plural). Although I can appreciate that he wanted to emphasize the sense in which the word ‘space’ was being used there, it\'s simply not correct there.

Page moves as such aren't kept in the history, but it seems likely that Urs did this last June, which is certainly when he added the term ‘topological space’ to the lede (and the redirect from the plural). Although I can appreciate that he wanted to emphasize the sense in which the word ‘space’ was being used there, it's simply not correct there.
- CommentRowNumber5.
- CommentAuthorTobyBartels
- CommentTimeAug 28th 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexI\'ve fixed up the lede, added a paragraph on complete topological spaces, and moved the page back to [[complete space]]. But we might still want to make [[complete topological space]] its own separate page, since it is somewhat a separate topic.

I've fixed up the lede, added a paragraph on complete topological spaces, and moved the page back to complete space. But we might still want to make complete topological space its own separate page, since it is somewhat a separate topic.
- CommentRowNumber6.
- CommentAuthorMike Shulman
- CommentTimeAug 28th 2015
- PermaLink
Author: Mike Shulman
Format: MarkdownItexThanks! [wikipedia's disambiguation page](https://en.wikipedia.org/wiki/Complete_topological_space) looks pretty good...

Thanks! wikipedia’s disambiguation page looks pretty good…
- CommentRowNumber7.
- CommentAuthorTobyBartels
- CommentTimeAug 29th 2015
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Author: TobyBartels
Format: MarkdownItexH\'m, it seems that I\'m remembering the ‘well established’ terminology wrong, in that it should be ‘topologically complete space’ rather than ‘complete topological space’. Everything at Wikipedia\'s disambiguation page is a special case of a Cauchy-complete Cauchy space, except for Čech completeness (which is still related). I\'d rather not copy that. On another note, I don\'t like using scare quotes for something that can be interpreted literally. A non-complete space really *does* have gaps, one for each point in its completion that\'s not in the original space.

H'm, it seems that I'm remembering the ‘well established’ terminology wrong, in that it should be ‘topologically complete space’ rather than ‘complete topological space’.

Everything at Wikipedia's disambiguation page is a special case of a Cauchy-complete Cauchy space, except for Čech completeness (which is still related). I'd rather not copy that.

On another note, I don't like using scare quotes for something that can be interpreted literally. A non-complete space really does have gaps, one for each point in its completion that's not in the original space.
- CommentRowNumber8.
- CommentAuthorTobyBartels
- CommentTimeAug 29th 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexThe terminology ‘topologically complete’ was originally meant to apply to metric spaces; the idea being that a given metric space might not be complete but might still be ‘topologically’ complete. It is a way to force completeness to become a topological property, specifically the topological property generated by the original (but not topological) property of completeness. That explains why the words come in a funny order. Also, I added some references, fixed the incomplete note about CTB vs compact, and expanded most sections.

The terminology ‘topologically complete’ was originally meant to apply to metric spaces; the idea being that a given metric space might not be complete but might still be ‘topologically’ complete. It is a way to force completeness to become a topological property, specifically the topological property generated by the original (but not topological) property of completeness. That explains why the words come in a funny order.

Also, I added some references, fixed the incomplete note about CTB vs compact, and expanded most sections.
- CommentRowNumber9.
- CommentAuthorTodd_Trimble
- CommentTimeAug 29th 2015
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Author: Todd_Trimble
Format: MarkdownItexAn intrinsically topological characterization of completely metrizable spaces is that they are Tychonoff spaces with a $\sigma$-locally finite base (Nagata-Smirnov), such that its canonical embedding into its Stone-Cech compactification is a countable intersection of open subspaces. Is there a similar intrinsic characterization of complete Cauchy spaces? I still feel there is some residual awkwardness with the terminology 'topologically complete space', but I suppose #7 is trying to tell me that '(Cauchy-)complete Cauchy space' is not adequate either. What is Čech completeness?

An intrinsically topological characterization of completely metrizable spaces is that they are Tychonoff spaces with a $\sigma$ -locally finite base (Nagata-Smirnov), such that its canonical embedding into its Stone-Cech compactification is a countable intersection of open subspaces. Is there a similar intrinsic characterization of complete Cauchy spaces?

I still feel there is some residual awkwardness with the terminology ’topologically complete space’, but I suppose #7 is trying to tell me that ’(Cauchy-)complete Cauchy space’ is not adequate either. What is Čech completeness?
- CommentRowNumber10.
- CommentAuthorTobyBartels
- CommentTimeAug 30th 2015
- (edited Aug 30th 2015)
- PermaLink
Author: TobyBartels
Format: MarkdownItex@Todd >Is there a similar intrinsic characterization of complete Cauchy spaces? If you mean intrinsic within the theory of Cauchy spaces, then we already have that; the usual concept of a complete space as one in which every Cauchy filter converges is already intrinsic to Cauchy spaces. If you mean a characterization of completely Cauchy-izable topological spaces, then I don\'t know of one. For the intermediate case of a completely uniformizable topological space, you can characterize these as the completely regular spaces whose fine uniformities are complete. This doesn\'t sound inherent, but since it refers to a uniformity that can be explicitly constructed, it can be stated without mentioning uniformities (or any other general notion of extra structure) at all. >I still feel there is some residual awkwardness with the terminology 'topologically complete space', but I suppose #7 is trying to tell me that '(Cauchy-)complete Cauchy space' is not adequate either. The terms ‘topologically complete space’ and ‘(Cauchy-)complete Cauchy space’ don\'t mean at all the same thing. (For example, the open unit interval in the real line is topologically complete but not Cauchy-complete.) Even comparing them depends heavily on context; the first requires you to form the underlying topological space (and then see if it is completely metrizable), while the second requires you to form the underlying Cauchy space (and then see if it is complete). And what this means depends on what structure you started with in the first place. >What is Čech completeness? This is described in the Arkhangel′skii reference that I added; it seems to be related to Baire category. In particular, every Čech-complete space is a Baire space, and a metrizable space is Čech-complete iff it\'s completely metrizable (aka topologically complete). Of course, these two facts don\'t explain why it matters, since you could simply speak of completely metrizable spaces instead, and these two facts would still be true.

@Todd

Is there a similar intrinsic characterization of complete Cauchy spaces?

If you mean intrinsic within the theory of Cauchy spaces, then we already have that; the usual concept of a complete space as one in which every Cauchy filter converges is already intrinsic to Cauchy spaces. If you mean a characterization of completely Cauchy-izable topological spaces, then I don't know of one.

For the intermediate case of a completely uniformizable topological space, you can characterize these as the completely regular spaces whose fine uniformities are complete. This doesn't sound inherent, but since it refers to a uniformity that can be explicitly constructed, it can be stated without mentioning uniformities (or any other general notion of extra structure) at all.

I still feel there is some residual awkwardness with the terminology ’topologically complete space’, but I suppose #7 is trying to tell me that ’(Cauchy-)complete Cauchy space’ is not adequate either.

The terms ‘topologically complete space’ and ‘(Cauchy-)complete Cauchy space’ don't mean at all the same thing. (For example, the open unit interval in the real line is topologically complete but not Cauchy-complete.) Even comparing them depends heavily on context; the first requires you to form the underlying topological space (and then see if it is completely metrizable), while the second requires you to form the underlying Cauchy space (and then see if it is complete). And what this means depends on what structure you started with in the first place.

What is Čech completeness?

This is described in the Arkhangel′skii reference that I added; it seems to be related to Baire category. In particular, every Čech-complete space is a Baire space, and a metrizable space is Čech-complete iff it's completely metrizable (aka topologically complete). Of course, these two facts don't explain why it matters, since you could simply speak of completely metrizable spaces instead, and these two facts would still be true.
- CommentRowNumber11.
- CommentAuthorTodd_Trimble
- CommentTimeAug 30th 2015
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Author: Todd_Trimble
Format: MarkdownItexI meant, and obviously should have said, "completely Cauchy-izable" in both places in #9. (Of course I already knew what you spelled out for me under the second citation in #10.) Thanks for pointing out Arkhangel′skii.

I meant, and obviously should have said, “completely Cauchy-izable” in both places in #9. (Of course I already knew what you spelled out for me under the second citation in #10.)

Thanks for pointing out Arkhangel′skii.
- CommentRowNumber12.
- CommentAuthorMike Shulman
- CommentTimeAug 30th 2015
- PermaLink
Author: Mike Shulman
Format: MarkdownItex> A non-complete space really does have gaps, one for each point in its completion that's not in the original space. But the word "gap" is not commonly given a precise mathematical definition. I don't understand why one needs a new confusing term for "completely metrizable".

A non-complete space really does have gaps, one for each point in its completion that’s not in the original space.

But the word “gap” is not commonly given a precise mathematical definition.

I don’t understand why one needs a new confusing term for “completely metrizable”.
- CommentRowNumber13.
- CommentAuthorTobyBartels
- CommentTimeAug 30th 2015
- (edited Aug 31st 2015)
- PermaLink
Author: TobyBartels
Format: MarkdownItexFirst of all, it\'s not a new term; it\'s an old one, and probably the older of the two. People referred to metric spaces as being topologically complete, a weaker condition that being complete, and you wouldn\'t call a metric space ‘metrizable’. Secondly, the two terms ‘completely metrizable’ and ‘topologically complete’ do *not* mean the same thing; for example, a uniform space (such as the open unit interval in the real line) may be topologically complete without being completely metrizable. Indeed, for a metric space, being completely metrizable just means being complete, because a metric space is already metrized. More abstractly, ‘completely metrizable’ applies to an object $X$ in a category $S$ equipped with a ‘forgetful’ functor $U\colon Met \to S$; then $X$ is completely metrizable iff there is a complete metric space $M$ such that $U(M)$ is isomorphic to $X$ in $S$. (The classical notion really assumes that $S$ and $U$ are strict and requires that $U(M) = X$. This is equivalent if $U$ is an isofibration.) On the other hand, ‘topologically complete’ applies to an object $X$ in a category $S$ equipped with a functor $V\colon S \to Top$; then $X$ is topologically complete iff $V(X)$ is completely metrizable (using the usual forgetful functor $Met \to Top$). If both concepts apply and the composite $Met \to S \to Top$ is the usual forgetful functor (or just naturally isomorphic to it), then a completely metrizable space must be topologically complete, but not conversely. Or more briefly, $X$ is topologically complete iff the underlying topological space of $X$ is completely metrizable, even if $X$ itself is not.

First of all, it's not a new term; it's an old one, and probably the older of the two. People referred to metric spaces as being topologically complete, a weaker condition that being complete, and you wouldn't call a metric space ‘metrizable’. Secondly, the two terms ‘completely metrizable’ and ‘topologically complete’ do not mean the same thing; for example, a uniform space (such as the open unit interval in the real line) may be topologically complete without being completely metrizable. Indeed, for a metric space, being completely metrizable just means being complete, because a metric space is already metrized.

More abstractly, ‘completely metrizable’ applies to an object $X$ in a category $S$ equipped with a ‘forgetful’ functor $U\colon Met \to S$ ; then $X$ is completely metrizable iff there is a complete metric space $M$ such that $U(M)$ is isomorphic to $X$ in $S$ . (The classical notion really assumes that $S$ and $U$ are strict and requires that $U(M) = X$ . This is equivalent if $U$ is an isofibration.) On the other hand, ‘topologically complete’ applies to an object $X$ in a category $S$ equipped with a functor $V\colon S \to Top$ ; then $X$ is topologically complete iff $V(X)$ is completely metrizable (using the usual forgetful functor $Met \to Top$ ). If both concepts apply and the composite $Met \to S \to Top$ is the usual forgetful functor (or just naturally isomorphic to it), then a completely metrizable space must be topologically complete, but not conversely.

Or more briefly, $X$ is topologically complete iff the underlying topological space of $X$ is completely metrizable, even if $X$ itself is not.
- CommentRowNumber14.
- CommentAuthorTobyBartels
- CommentTimeAug 30th 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexThis isn\'t to say that ‘topologically complete’ is a good term. It\'s not; it\'s too vague. But the adverb ‘topologically’ isn\'t at fault; it does real work. A more systematic term would be ‘topologically completely metrizable’. That is, think about it topologically (take the underlying topological space), and only then consider whether it\'s completely metrizable. One could also say ‘topologically completely uniformizable’ (a weaker condition), ‘uniformly completely metrizable’ (a stronger condition), etc.

This isn't to say that ‘topologically complete’ is a good term. It's not; it's too vague. But the adverb ‘topologically’ isn't at fault; it does real work.

A more systematic term would be ‘topologically completely metrizable’. That is, think about it topologically (take the underlying topological space), and only then consider whether it's completely metrizable. One could also say ‘topologically completely uniformizable’ (a weaker condition), ‘uniformly completely metrizable’ (a stronger condition), etc.
- CommentRowNumber15.
- CommentAuthorTodd_Trimble
- CommentTimeAug 30th 2015
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Author: Todd_Trimble
Format: MarkdownItex> More abstractly, ‘completely metrizable’ applies to an object $X$ in a category $S$ equipped with a ‘forgetful’ functor $U\colon Met \to S$; then $X$ is completely metrizable iff there is a metric space $M$ such that $U(M)$ is isomorphic to $X$ in $S$. You meant *complete* metric space $M$. > Indeed, for a metric space, being completely metrizable just means being complete, because a metric space is already metrized. In reality, if I heard someone say that some metric space is completely metrizable, I'm almost certain I wouldn't think he just means it's complete! Actually it's hard for me to believe that anyone in reality would deploy this terminology according to the definition above and still be asserting something that is correct! That is to say, the terminology "completely metrizable metric space", while logically defensible if one follows the (amended) definition above, would in reality almost certainly be very confusing if the speaker really did have in mind the particular case where $S = Met$ and $U$ is the identity functor. (I don't suppose you have a citation for that particular usage?) In any case, if I *knew* that was really meant, I'd still wonder why he (I'll say it's a 'he'!) is acting all weird and didn't just say "complete metric space". To me it's just as weird as "metrizable metric space" (which you said someone *wouldn't* say). If I *didn't* know that, I'd probably assume the speaker means that one is able to describe its topology as that given by a complete metric, not necessarily the assigned metric. For example someone might say the metric space of irrationals under the Euclidean distance metric is not complete but it is completely metrizable (here considering the Euclidean metric merely as a means to the end of specifying the topology). I'm sure I wouldn't even blink an eye. All this is to say that I think "completely metrizable" by itself can safely be taken as meaning by default "topologically completely metrizable", i.e., referring to the forgetful functor $Met \to Top$. Otherwise someone could say e.g. "uniformly completely metrizable", or the sufficiently bizarre mathematician could say "metrically completely metrizable", etc., if something other than the default meaning is intended. > Or more briefly, $X$ is topologically complete iff the underlying topological space of $X$ is completely metrizable, even if $X$ itself is not. Of course only under the understanding that one is referring to the underlying functor $Met \to Top$. If I understand what is being said in #14, then "topologically complete" could also mean something else, like topologically completely uniformizable.

More abstractly, ‘completely metrizable’ applies to an object $X$ in a category $S$ equipped with a ‘forgetful’ functor $U\colon Met \to S$ ; then $X$ is completely metrizable iff there is a metric space $M$ such that $U(M)$ is isomorphic to $X$ in $S$ .

You meant complete metric space $M$ .

Indeed, for a metric space, being completely metrizable just means being complete, because a metric space is already metrized.

In reality, if I heard someone say that some metric space is completely metrizable, I’m almost certain I wouldn’t think he just means it’s complete! Actually it’s hard for me to believe that anyone in reality would deploy this terminology according to the definition above and still be asserting something that is correct!

That is to say, the terminology “completely metrizable metric space”, while logically defensible if one follows the (amended) definition above, would in reality almost certainly be very confusing if the speaker really did have in mind the particular case where $S = Met$ and $U$ is the identity functor. (I don’t suppose you have a citation for that particular usage?) In any case, if I knew that was really meant, I’d still wonder why he (I’ll say it’s a ’he’!) is acting all weird and didn’t just say “complete metric space”. To me it’s just as weird as “metrizable metric space” (which you said someone wouldn’t say).

If I didn’t know that, I’d probably assume the speaker means that one is able to describe its topology as that given by a complete metric, not necessarily the assigned metric. For example someone might say the metric space of irrationals under the Euclidean distance metric is not complete but it is completely metrizable (here considering the Euclidean metric merely as a means to the end of specifying the topology). I’m sure I wouldn’t even blink an eye.

All this is to say that I think “completely metrizable” by itself can safely be taken as meaning by default “topologically completely metrizable”, i.e., referring to the forgetful functor $Met \to Top$ . Otherwise someone could say e.g. “uniformly completely metrizable”, or the sufficiently bizarre mathematician could say “metrically completely metrizable”, etc., if something other than the default meaning is intended.

Or more briefly, $X$ is topologically complete iff the underlying topological space of $X$ is completely metrizable, even if $X$ itself is not.

Of course only under the understanding that one is referring to the underlying functor $Met \to Top$ . If I understand what is being said in #14, then “topologically complete” could also mean something else, like topologically completely uniformizable.
- CommentRowNumber16.
- CommentAuthorTobyBartels
- CommentTimeAug 30th 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItex>You meant *complete* metric space $M$. Yes, sorry! Fixed. >if I heard someone say that some metric space is completely metrizable I would think that they had misspoken or that I had misunderstood. Nobody would actually say that, unless having some highly abstract discussion such as I was having in #13. >For example someone might say the metric space of irrationals under the Euclidean distance metric is not complete but it is completely metrizable (here considering the Euclidean metric merely as a means to the end of specifying the topology). I'm sure I wouldn't even blink an eye. I would ask for clarification. And if they really meant what you\'d said, then I\'d wonder why they didn\'t say ‘topologically complete’ instead, since that\'s the usual terminology IME. However, if they said ‘the space of irrationals’ rather than ‘the metric space of irrationals under the Euclidean distance metric’, then I probably wouldn\'t blink, but I would assume that this was somebody for whom ‘space’ means ‘topological space’ by default. (And maybe also somebody for whom ‘irrational’ means ‘real irrational’ by default, for which I blame the textbooks that I have to teach out of, although the truth of these statements doesn\'t heavily depend on that.) >All this is to say that I think "completely metrizable" by itself can safely be taken as meaning by default "topologically completely metrizable", i.e., referring to the forgetful functor $Met \to Top$. So if somebody said ‘completely metrizable uniform space’, would you assume that they meant to include ${]{0,1}[}$ as an example? I would not! >"topologically complete" could also mean something else, like topologically completely uniformizable Wikipedia claims that this term has been used with that meaning, but it\'s certainly not the meaning that I\'m familiar with.

You meant complete metric space $M$ .

Yes, sorry! Fixed.

if I heard someone say that some metric space is completely metrizable

I would think that they had misspoken or that I had misunderstood. Nobody would actually say that, unless having some highly abstract discussion such as I was having in #13.

For example someone might say the metric space of irrationals under the Euclidean distance metric is not complete but it is completely metrizable (here considering the Euclidean metric merely as a means to the end of specifying the topology). I’m sure I wouldn’t even blink an eye.

I would ask for clarification. And if they really meant what you'd said, then I'd wonder why they didn't say ‘topologically complete’ instead, since that's the usual terminology IME. However, if they said ‘the space of irrationals’ rather than ‘the metric space of irrationals under the Euclidean distance metric’, then I probably wouldn't blink, but I would assume that this was somebody for whom ‘space’ means ‘topological space’ by default. (And maybe also somebody for whom ‘irrational’ means ‘real irrational’ by default, for which I blame the textbooks that I have to teach out of, although the truth of these statements doesn't heavily depend on that.)

All this is to say that I think “completely metrizable” by itself can safely be taken as meaning by default “topologically completely metrizable”, i.e., referring to the forgetful functor $Met \to Top$ .

So if somebody said ‘completely metrizable uniform space’, would you assume that they meant to include ${]{0,1}[}$ as an example? I would not!

“topologically complete” could also mean something else, like topologically completely uniformizable

Wikipedia claims that this term has been used with that meaning, but it's certainly not the meaning that I'm familiar with.
- CommentRowNumber17.
- CommentAuthorTodd_Trimble
- CommentTimeAug 30th 2015
- (edited Aug 30th 2015)
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexToby, I went ahead and tried my hand at editing at [[complete space]]. I hope I did not make a complete hash of it, and hope also you don't mind. I tried to follow what's been said in this discussion, mainly with a view of making matters clear to myself (I had found the prior revision somewhat confusing). Note particularly that I changed the wording in the Idea section slightly, and tried to inject some of the formal or abstract meanings you introduced above in this discussion. > So if somebody said ‘completely metrizable uniform space’, would you assume that they meant to include ${]{0,1}[}$ as an example? I would not! Sorry, which uniformity do you mean? If you mean the one coming from the usual Euclidean metric, then no I wouldn't either. So, when I said "completely metrizable" *by itself*, I did mean referring to topological spaces by default; by saying "completely metrizable *uniform* space", you declare that the default meaning is no longer in effect. Am I making sense now? :-) **Edit:** Whoops! Sorry, I'm a little slow. A metric space is complete iff its underlying uniform space is complete, so now I guess I see better what your objection is to that last example. There's a spot in [[complete space]] which reflects that I didn't think that particular point through, but I'll wait til you give it a look-over and report back.

Toby, I went ahead and tried my hand at editing at complete space. I hope I did not make a complete hash of it, and hope also you don’t mind. I tried to follow what’s been said in this discussion, mainly with a view of making matters clear to myself (I had found the prior revision somewhat confusing). Note particularly that I changed the wording in the Idea section slightly, and tried to inject some of the formal or abstract meanings you introduced above in this discussion.

So if somebody said ‘completely metrizable uniform space’, would you assume that they meant to include ${]{0,1}[}$ as an example? I would not!

Sorry, which uniformity do you mean? If you mean the one coming from the usual Euclidean metric, then no I wouldn’t either. So, when I said “completely metrizable” by itself, I did mean referring to topological spaces by default; by saying “completely metrizable uniform space”, you declare that the default meaning is no longer in effect. Am I making sense now? :-)

Edit: Whoops! Sorry, I’m a little slow. A metric space is complete iff its underlying uniform space is complete, so now I guess I see better what your objection is to that last example. There’s a spot in complete space which reflects that I didn’t think that particular point through, but I’ll wait til you give it a look-over and report back.
- CommentRowNumber18.
- CommentAuthorTodd_Trimble
- CommentTimeAug 30th 2015
- PermaLink
Author: Todd_Trimble
Format: MarkdownItex> I'd wonder why they didn't say ‘topologically complete’ instead, since that's the usual terminology IME. IME = "in my experience"? Your experience is much different from mine. I'd never even heard "topologically complete" before this discussion!

I’d wonder why they didn’t say ‘topologically complete’ instead, since that’s the usual terminology IME.

IME = “in my experience”? Your experience is much different from mine. I’d never even heard “topologically complete” before this discussion!
- CommentRowNumber19.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItex>by saying "completely metrizable *uniform* space", you declare that the default meaning is no longer in effect Inasmuch as for most people the default meaning of ‘space’ is topological space, then yes, by sticking ‘uniform’ in there, I change the default. But when I stuck ‘metric’ in there, you claimed that this did *not* change the default. Just as ${]{0,1}[}$ (with its usual uniformity) is not a completely metrizable uniform space, so it is also (with its usual metric) not a completely metrizable metric space, even though (with its usual topology) it is a completely metrizable topological space. Actually, if somebody said ‘completely metrizable metric space’ *and* I had good reason to believe that there was no mistake (they did not misspeak and I did not misunderstand) *and* this was somebody who\'s probably not speaking at a high level of abstraction like I was doing in \#13, *then* I would probably assume that they meant topologically completely metrizable. This on the grounds that topological spaces are so often taken as the default. But if I were editing the manuscript, then I would advise heavily against it!

by saying “completely metrizable uniform space”, you declare that the default meaning is no longer in effect

Inasmuch as for most people the default meaning of ‘space’ is topological space, then yes, by sticking ‘uniform’ in there, I change the default. But when I stuck ‘metric’ in there, you claimed that this did not change the default. Just as ${]{0,1}[}$ (with its usual uniformity) is not a completely metrizable uniform space, so it is also (with its usual metric) not a completely metrizable metric space, even though (with its usual topology) it is a completely metrizable topological space.

Actually, if somebody said ‘completely metrizable metric space’ and I had good reason to believe that there was no mistake (they did not misspeak and I did not misunderstand) and this was somebody who's probably not speaking at a high level of abstraction like I was doing in #13, then I would probably assume that they meant topologically completely metrizable. This on the grounds that topological spaces are so often taken as the default. But if I were editing the manuscript, then I would advise heavily against it!
- CommentRowNumber20.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- (edited Aug 31st 2015)
- PermaLink
Author: TobyBartels
Format: MarkdownItexI find the paragraph ending with ‘another synonym for this, still in the uniform space context, is Dieudonné-complete.’ rather confusing. Regarding this last clause in particular, there are always *two* contexts for this stuff; for Dieudonné completeness (which is topological complete uniformizability in the language of #14), the contexts are $Top$ and $Unif$, not just $Unif$. (For topological completeness, that is topological complete metrizability, the contexts are $Top$ and $Met$; for uniform complete metrizability, they are $Unif$ and $Met$; etc.) And here: >then analogously we can refer to an object $X$ of $\mathcal{S}$ as being "$\mathcal{T}$-ly $\mathcal{S}$-complete" This is not the most general case, and it doesn\'t even include the case of a topologically complete uniform space. Besides the two contexts that I mentioned above, there is whatever context the object $X$ is given in. So if you have some standard ‘forgetful’ functors $\mathcal{R} \to \mathcal{S} \to \mathcal{T}$ (and also $\mathcal{R} \to Cauch$, so that we know when an $\mathcal{R}$-object is ‘complete’), then an $\mathcal{S}$-object $X$ is $\mathcal{T}$-ly $\mathcal{R}$-complete (or $\mathcal{T}$-ly completely $\mathcal{R}$-able) if there is some complete $\mathcal{R}$-object $Y$ such that the underlying $\mathcal{T}$-objects of $X$ and $Y$ are isomorphic. Of course, I can edit this. But first I want to discuss something here: I think that dealing with all of this stuff about topological completeness distracts from the main idea of [[complete space]], and I\'d like to spin off [[topologically complete space]] as a separate page.

I find the paragraph ending with ‘another synonym for this, still in the uniform space context, is Dieudonné-complete.’ rather confusing. Regarding this last clause in particular, there are always two contexts for this stuff; for Dieudonné completeness (which is topological complete uniformizability in the language of #14), the contexts are $Top$ and $Unif$ , not just $Unif$ . (For topological completeness, that is topological complete metrizability, the contexts are $Top$ and $Met$ ; for uniform complete metrizability, they are $Unif$ and $Met$ ; etc.)

And here:

then analogously we can refer to an object $X$ of $\mathcal{S}$ as being “ $\mathcal{T}$ -ly $\mathcal{S}$ -complete”

This is not the most general case, and it doesn't even include the case of a topologically complete uniform space. Besides the two contexts that I mentioned above, there is whatever context the object $X$ is given in. So if you have some standard ‘forgetful’ functors $\mathcal{R} \to \mathcal{S} \to \mathcal{T}$ (and also $\mathcal{R} \to Cauch$ , so that we know when an $\mathcal{R}$ -object is ‘complete’), then an $\mathcal{S}$ -object $X$ is $\mathcal{T}$ -ly $\mathcal{R}$ -complete (or $\mathcal{T}$ -ly completely $\mathcal{R}$ -able) if there is some complete $\mathcal{R}$ -object $Y$ such that the underlying $\mathcal{T}$ -objects of $X$ and $Y$ are isomorphic.

Of course, I can edit this. But first I want to discuss something here: I think that dealing with all of this stuff about topological completeness distracts from the main idea of complete space, and I'd like to spin off topologically complete space as a separate page.
- CommentRowNumber21.
- CommentAuthorTodd_Trimble
- CommentTimeAug 31st 2015
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Author: Todd_Trimble
Format: MarkdownItexToby, if you could edit the paragraph mentioned in #20, that would be great; sorry for any trouble. I hope the rest was accurate.

Toby, if you could edit the paragraph mentioned in #20, that would be great; sorry for any trouble. I hope the rest was accurate.
- CommentRowNumber22.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- (edited Aug 31st 2015)
- PermaLink
Author: TobyBartels
Format: MarkdownItexI just added to \#20, not realizing that you were reading it at the same time. ETA: It\'s good that you\'re writing all of this, even if I want to change some of it afterward. :–)

I just added to #20, not realizing that you were reading it at the same time. ETA: It's good that you're writing all of this, even if I want to change some of it afterward. :–)
- CommentRowNumber23.
- CommentAuthorTodd_Trimble
- CommentTimeAug 31st 2015
- PermaLink
Author: Todd_Trimble
Format: MarkdownItex> But first I want to discuss something here: I think that dealing with all of this stuff about topological completeness distracts from the main idea of complete space You're probably right. > I'd like to spin off topologically complete space as a separate page. Sure, go for it! Sounds good to me.

But first I want to discuss something here: I think that dealing with all of this stuff about topological completeness distracts from the main idea of complete space

You’re probably right.

I’d like to spin off topologically complete space as a separate page.

Sure, go for it! Sounds good to me.
- CommentRowNumber24.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexI did the split and slightly rearranged the sections of [[complete space]], adding a brief remark about Cech-completeness as well. I haven\'t yet edited anything that I imported to [[topologically complete space]].

I did the split and slightly rearranged the sections of complete space, adding a brief remark about Cech-completeness as well. I haven't yet edited anything that I imported to topologically complete space.
- CommentRowNumber25.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexNow I have edited [[topologically complete space]]. I\'m sorry, but very little of your text survived, Todd. However, you can edit it some more now if you wish!

Now I have edited topologically complete space. I'm sorry, but very little of your text survived, Todd. However, you can edit it some more now if you wish!
- CommentRowNumber26.
- CommentAuthorTodd_Trimble
- CommentTimeAug 31st 2015
- PermaLink
Author: Todd_Trimble
Format: MarkdownItex> very little of your text survived That in itself is not a problem. However, l am confused all over again. Probably by a number of things. As it's late here and I'm working on something else at the moment, I'll have to take it up again later (i.e., ask questions here, unless I get clear on it by myself). Anyway, thanks for doing this.

very little of your text survived

That in itself is not a problem. However, l am confused all over again. Probably by a number of things. As it’s late here and I’m working on something else at the moment, I’ll have to take it up again later (i.e., ask questions here, unless I get clear on it by myself). Anyway, thanks for doing this.
- CommentRowNumber27.
- CommentAuthorMike Shulman
- CommentTimeAug 31st 2015
- PermaLink
Author: Mike Shulman
Format: MarkdownItexMy response to #13 would have been the same as Todd's #15, but now I also see that the case of uniform spaces (and more generally things other than topological spaces) means we need a different word. But do we have to perpetuate bad terminology? Can we just say "topologically completely metrizable"?

My response to #13 would have been the same as Todd’s #15, but now I also see that the case of uniform spaces (and more generally things other than topological spaces) means we need a different word. But do we have to perpetuate bad terminology? Can we just say “topologically completely metrizable”?
- CommentRowNumber28.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexSure, but I invented that phrasing in comment #14. Hopefully it\'s sufficiently self-explanatory, but it\'s not in the literature.

Sure, but I invented that phrasing in comment #14. Hopefully it's sufficiently self-explanatory, but it's not in the literature.
- CommentRowNumber29.
- CommentAuthorMike Shulman
- CommentTimeAug 31st 2015
- PermaLink
Author: Mike Shulman
Format: MarkdownItexIn other places on the lab we've made an effort to improve existing terminology.

In other places on the lab we’ve made an effort to improve existing terminology.
- CommentRowNumber30.
- CommentAuthorTobyBartels
- CommentTimeAug 31st 2015
- PermaLink
Author: TobyBartels
Format: MarkdownItexWell, yes, and I hope that that\'s what I\'m doing at [[topologically complete space]]. Nevertheless, I\'m also recording the previous terminology.

Well, yes, and I hope that that's what I'm doing at topologically complete space. Nevertheless, I'm also recording the previous terminology.

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