Added fact that relational beta-modules and topological spaces are only equivalent to each other if the ultrafilter principle is true.

]]>I have made the `!include`

of the list of references come out by adding vertical whitespace to it

added requirement that the unions be $U$-small in dependent type theory, as one cannot quantify over arbitrary types, only the $U$-small types relative to a universe $U$

]]>Sorry about that, I forgot to write what i did in the summary for this article. The only thing I did here was add a link to the $\sigma$-topological space article in the Related Concepts section.

]]>Oh, sorry about that, never mind. I see you did that on *other* articles (which I hadn’t seen while I was looking around at what you did), just not this article.

Well, well, well. Just goes to show me, I guess.

Guest, could I ask you to make notes here at the nForum of your nLab edits, as the little window that opens up when you edit invites you to do? You just enter in a brief description of what you did, and hit Submit.

]]>Sure, that name would naturally suggest itself, but googling that term doesn’t seem to turn up anything of direct relevance to your question. My guess is that’s probably not because no one has ever considered it, but rather because it’s never led to anything significant.

(Especially in the old days of the nLab, people would experiment and noodle around a little more than I see now. Witness the existence of the Boolean rig article, seemingly a side curiosity.)

]]>Maybe $\sigma$-topological space? $\sigma$ is used to indicate countable unions or joins in various other topological and measure theoretic structures, such as $\sigma$-locale, $\sigma$-frame, $\sigma$-algebra, $\sigma$-complete lattice, $\sigma$-continuous valuation, et cetera.

]]>Re #14: not that I can think of at the moment, but in many cases, just having closure under countable unions is enough to guarantee closure under arbitrary unions. These fall under the ambit of countability axioms (first countable space, second countable space, which I am liable to mix up which is which).

]]>Did some work on the definitions section, which was a little rough in places. Saving for the moment.

]]>Is there a name for a topological space where the open sets are only closed under countable unions rather than arbitrary unions?

]]>added more hyperlinks to technical terms in the first few paragraphs.

Inserted a sentence early on which highlights that most notions of spaces in math have underlying topological spaces (the sentence could easily be expanded further).

also made some mild edits to the wording and formatting in the first few paragraphs

]]>I made some of this history more precise, and did some smoothing of the narrative under Variations. I will add references later.

]]>Added sentence

]]>Hausdorff’s definition originally contained the $T_2$-separation axiom, which was removed by Kazimierz Kuratowski in 1922 resulting in the current defintion of topological space.

Replaced

The definition of topological space was a matter of some debate, especially about 100 years ago. Our definition is due to Bourbaki, so may be called

Bourbaki spaces.

with

The definition of topological space used in this article involving neighbourhoods was first developed by Felix Hausdorff in 1914 in his seminal text on set theory and topology, *Fundamentals of Set Theory* (*Grundzüge der Mengenlehre*).

The 1914 definition is for Hausdorff topological spaces. Kuratowski was the first one to drop the Hausdorff axiom.

]]>The sentence about calling these structures “Bourbaki spaces” was added by Toby Bartels in the 2nd revision of the page:

]]>Currently we have the sentence, “The definition of topological space was a matter of some debate, especially about 100 years ago. Our definition is due to Bourbaki, so may be called Bourbaki spaces.”

Is it? Due to Bourbaki? And not, say, Kuratowski in 1922, predating even the existence of Bourbaki by more than a decade? Besides that, I have never heard anyone refer to topological spaces as “Bourbaki spaces”.

]]>I did some rewriting of the Idea section, keeping (or in some cases slightly rearranging) phrases with words like “cohesion” and “cohere”, but eliminating some others which I thought could be confusing, per discussion above.

]]>(Over three years later, but first time seeing #2.) Honestly, I don’t think that word is likely to be helpful; indeed, I think it’s likely going to confuse some people such as mdiamond. In general, the Idea section looks simultaneously overworked and full of fluff.

]]>added link to ionad

]]>I tried to prettify the entry *topological space* a bit more:

made an attempt at adding an Idea-section (feel free to work on that, it’s just a quick idea motivated more from the desire to have such a section at all than from an attempt to do it any justice).

collected the three Definition-sections to subsections of a single Definition-section

polished and expanded the Standard definition section.