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There was already a discussion of ends in the topological sense at proper homotopy. (I had never seen the term hemicompact before. I knew of $\sigma$-compact which is almost the same.)
Yes, I wrote end compactification based off of Wikipedia. I still want to convince myself that this is all self-consistent.
At proper homotopy theory, the definition uses the closures of compact subspaces rather than the compact subspaces themselves. It does seem to be a little easier to define the concept using these. Are there examples to show why one way is better than the other? Or are they secretly equivalent?
I quote from Wikipedia
Every hemicompact space is σ-compact. The converse, however, is not true; for example, the space of rationals, with the usual topology, is σ-compact but not hemicompact.
Yes, I noticed that. I don't know if that was in answer to my question
Are there examples to show why one way is better than the other? Or are they secretly equivalent?
but that's not what I was asking about.
I was asking about using compact subpsaces versus the closures of compact subspaces in the definition of ‘end’.
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