Talk:Subspace topology

Redirects to chapters
Why do redirects to chapters of an article not work?? Revolver 05:57, 12 Dec 2004 (UTC)

Rename article
I recently created subspace topology and direct sum topology. The article names were choosen in order to fit in with quotient topology and product topology. But now I think it would be better to rename all articles to product (topology) or topological product in order to put the focus on the construction and not on the topology. Any opinions ? MathMartin 20:29, 3 Mar 2005 (UTC)

I vote for -- Fropuff 01:57, 2005 Mar 4 (UTC)
 * subspace (topology)
 * quotient space
 * product space
 * disjoint union (topology)


 * Why didn't you write quotient space (topology) and product space (topology) to make the naming consistent ? Otherwise I like the names. MathMartin 13:43, 4 Mar 2005 (UTC)

As long as there isn't a conflict we should stick to the shorter names. It makes linking easier when writing other articles. I think striving for 100% naming consistency within Wikipedia is a futile task. -- Fropuff 15:26, 2005 Mar 4 (UTC)


 * Linking can be made easier be adding redirects. I do not aim for 100% naming consistency within Wikipedia but just for consistency regarding the main article titles for closely related articles. MathMartin 15:52, 4 Mar 2005 (UTC)

Notation
On an unrelated note: I suppose its a matter of personal preference, but I think the underline notation $$\underline{X}:=(X,\tau)$$ is distracting. It may be an abuse of notation, but it is a perfectly standard practice to call X a topological space with the topology being understood. -- Fropuff 02:28, 2005 Mar 4 (UTC)


 * I know that my underline notation is not standard but I think it is very confusing to just call X a topological space with the topology being understood. I changed the notation by omitting the name of the topological space and just writing the ordered tuple (X,&tau;).
 * I think it is important to distinguish clearly between the topological space (X,&tau;) and the underlying set X when describing constructions involving several topological spaces. MathMartin 13:43, 4 Mar 2005 (UTC)

I think its fine to be clear in the definition, but afterwards revert to the more casual notation. After all, there is little ambiguity: we are only talking about one topology on X and one topology on S. -- Fropuff 15:26, 2005 Mar 4 (UTC)


 * If you are referring to the Properties section and you think you can make the notation clearer then give it a try. But I doubt it is possible.MathMartin 15:52, 4 Mar 2005 (UTC)

I think
 * If $$S$$ is open in $$X$$ we call $$S$$ an open subspace of $$X$$

is harder to understand and more ambiguous than the slighly more verbose
 * If $$S$$ is open in $$(X,\tau)$$ we call $$(S,\tau_S)$$ an open subspace of $$(X,\tau)$$.

MathMartin 12:55, 5 Mar 2005 (UTC)

That's probably true. We should maybe also state that a subspace is open (closed) iff the inclusion map is an open (closed) map from $$(S,\tau_S)$$ to $$(X,\tau)$$. -- Fropuff 15:22, 2005 Mar 5 (UTC)

Error?
To the author: You say that $$\{0\} \in \mathbb{Q}$$ is not open so $$\mathbb{Q}$$ does not have the discrete topology. Isn't this incorrect? If we use the discrete topology on $$\mathbb{R}$$ then $$\{ 0 \} \in \mathbb{R}$$ is open. So $$\mathbb{Q} \cap \{0\} = \{0\} \in \tau_S$$ should be open as well. Am I missing something? Jtabbsvt 06:17, 24 April 2006 (UTC)


 * Yes, but R does not have the discrete topology, it has the Euclidean one. So {0} is not open is R or Q. -- Fropuff 17:00, 24 April 2006 (UTC)

I suppose the wording was what confused me. R "doesn't have" the discrete topology in your particular example, but you can certainly bestow the discrete topology upon R. Doing so would induce the discrete topology on Q. So it's not that R or Q "doesn't have" the discrete topology, but rather that the discrete topology on Q can only be induced from the discrete topology on R, and by contrast, it can't be induced by, say, the Euclidean topology on R.

My main point is that there are many ways to choose a topology for R (though admittedly the Euclidean topology is the usual one), so for the purpose of the example (which should be easily understood by the types of people that are familiarizing themselves with this material), it may be more useful to point out that you're using the Euclidean metric in the example.Jtabbsvt 02:20, 26 April 2006 (UTC)