Talk:Topological property

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We already have a huge list of topological properties at Topological space#Classification of topological spaces. Perhaps we should move that list here instead. -- Fropuff 00:58, 2005 Mar 6 (UTC)

I think it should be moved here. topological space is too long anyway. Note that I also created uniform properties some time ago as a parallel page discussing properties for uniform spaces. So on this page only properties preserved by continuous functions should be discussed. MathMartin 12:01, 6 Mar 2005 (UTC)

[edit] Is open set a topological property

Good, I agree, let's move it here. BTW, the first four properties you have on the list (open set, interior, neighborhood, and limit point) aren't really topological properties in the usual sense of the word. Topological properties are properties of spaces invariant under homeomorphism (okay, every topological space is open in itself, but that is rather tautological; or one could say that the set of open sets—the topology— is a topological property but again, that's pure tautology). -- Fropuff 17:05, 2005 Mar 6 (UTC)

Wittgenstein said all mathematics is tautology, and I share his opinion. If the homeomorphism maps the space onto itself and the topology is defined using the closure operator it is not tautological (at least not in my opinion) that open sets stay open under the homeomorphism. Or consider someone who does not know the definition of continuity in terms of open sets and is just looking for the properties studied in topology to get a feeling for the subject. My point is properties which you consider tautological might be suprising for someone else, depending on the direction he is coming from. Of course a balance has to be reached and it would certainly be overkill to explain why open sets are topological properties, but a short link certainly makes the article easier to understand.

If the article grows longer we can always put the tautological topological properties in a section called ==Basic properties==.

MathMartin 17:30, 6 Mar 2005 (UTC)

To me (and many others I suspect) the presence of the term open set looks very strange on a list of topological properties. I think we should either remove it or explain in what sense open sets are to be regarded as topological properties. -- Fropuff 17:46, 2005 Mar 6 (UTC)

Ok perhaps it should be explained better. I meant to say a homemorphism maps open sets to open sets. I can agree that this is a basic property of a topological space but I do not think it is strange to include it here. Perhaps someone else can give his opinion. MathMartin 18:03, 6 Mar 2005 (UTC)

I now have a better argument as to why openness should not be listed as a topological property. There are topological spaces (such as Hilbert spaces) where open sets are homeomorphic to non-open sets. Invariance of domain says that this can't happen in Euclidean space, but it does happen elsewhere. -- Fropuff 18:15, 3 October 2005 (UTC)

I do not understand. Can you provide more details ? MathMartin 18:32, 3 October 2005 (UTC)

See the book by Willard (General Topology, 1970). The example he provides (problem 18B) is that Hilbert space is isometric to a nowhere dense subset of itself. -- Fropuff 18:37, 3 October 2005 (UTC)

Let me phrase it a different way. It is certainly possible that a open subset A of a space X has a homeomorphic copy in a space Y that is not open in Y. For example, take A to be a single point of a discrete space, and take Y to be the real line. Any single point subspace of Y is homeomorphic to A even though it is not open in Y. -- Fropuff 19:06, 3 October 2005 (UTC)

Thanks for the explanation, but I still do not understand. If you have a homeomorphism from a single point subspace U of Y to A then you have to consider the subspace topology on U induced by Y in which U is open (and closed). MathMartin 19:20, 3 October 2005 (UTC)

It depends on what you mean when you say "being an open set is a topological property". There are a variety of possible meanings. One possible meaning is

• (O1) A space A can be embedded as an open subspace of another space.

Since every space is an open subset of itself, all topological spaces have the property O1. The statement O1 is then a topological property, but in a very trivial and uninteresting way. It can't distinguish between anything. A much more interesting problem is to consider the subsets of a given topological space X and ask whether the following statement is a topological property:

• (O2) A subspace A is open in X

My point is that the statement (O2) need not be a topological property of subspaces of X. There can be open subsets of X that are homeomorphic to non-open subsets of X. This is what Willard means when he says that openness is not a topological property (or even a metric property). Note that statement (O2) is a topological property of subspaces of Rⁿ by invariance of domain as I mentioned earlier. -- Fropuff 20:20, 3 October 2005 (UTC)