Talk:Closed extension topology

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Sorry, but this article doesn't make sense as it stands. For a start, shouldn't conditions (1) and (2) start with "p ∉ S" and "p ∈ S" rather than "p ∉ X" and "p ∈ X"? Also, don't you need X to be a topological space to start with, rather than just a set? (or something along those lines?) Dmharvey 02:36, 14 June 2006 (UTC)

(1) still reads bad if you can think of an easier way to say that.... More importantly I need to go through the whole particular point thing since most of them actually still hold for this case (except where I use discrete).

BTW I didn't understand what symbol changes you wanted. If you want make them here so I get some idea. jbolden1517^Talk 03:05, 14 June 2006 (UTC)

I'll get to some of those things soon. (Maybe tomorrow). Dmharvey 03:11, 14 June 2006 (UTC)

[edit] encyclopaedic style

The article is part of an encyclopaedia, not a textbook. It should start with something like

In mathematics, the closed extension topology is a construction involving topological spaces.

Also, it should give some indication of the relevance of the concept. Right now I'm not so sure what it's useful for, apart from perhaps constructing pathological spaces.

The problem is I'm not exactly sure myself. I'm sort of doing a personal wikiproject in creating articles for all counter examples (filling in the list) at Counterexamples in Topology. This is one of them, but its an aside not an article. Even when I was a mathematician I was never a topologist. You grabbed this much sooner than I had expected (I guess because of the stub). I was going to do something this weekend to tie this closer to particular point.

But the reason you don't know about the space is because it isn't particularly useful. I'm just working my way down the list. jbolden1517^Talk 11:06, 14 June 2006 (UTC)

OK, fair enough. Maybe I'll try making up something myself. :-) Dmharvey 12:49, 14 June 2006 (UTC)

[edit] primary definition?

It feels to me like the definition that you give in terms of "start with Y, add a new point p, to get a topology on Y ∪ p" is more natural and easier to understand than the current primary definition given in the article. Which way does the book do first? (that's a book I really should have read by now, but somehow never quite made it :-) Dmharvey 03:23, 14 June 2006 (UTC)

That's me not the book. I or you can flip it. And you definitely should buy the book (its $10), it is terrific; best advanced point set book I've ever seen. jbolden1517^Talk 11:09, 14 June 2006 (UTC)

[edit] problems with definition

I've looked at this more closely now and I'm very confused.

The first definition is not even a topology. It doesn't include the whole space as a closed set. When you say "[by] defining the closed sets on Y* to be the closed sets on Y", do you really mean "defining the closed sets of Y* to be those sets whose intersections with Y are closed in Y"? Or do you mean: "throw in all closed sets of Y, and then throw in Y* as well"?

The second definition is very strange when I think about it. It seems to be saying that S being open doesn't depend on whether p is in S at all; it only depends on whether S ∩ (X\{p}) is open. I suppose that could be what is intended, but it seems strange to me. It doesn't feel at all like the one-point compactification.

Also, I can't seem to make this match up with the particular point topology as you say it is. For the closed extension topology, it's like the topology doesn't care whether p belongs to the set or not; whereas for the particular point topology, the topology only cares whether p belongs.

Some clarification would be appreciated. Dmharvey 13:45, 14 June 2006 (UTC)