Is there a way to see whether a topological space $\Omega$ does not allow retractions $r: \Omega \mapsto B$, with $B$ a given subspace of $\Omega$ ?
In other words: when is a space not retractable to a given subspace $B$ ?
3
-
6$\begingroup$ Can you put quantifiers in your question? Are you asking for spaces such that there exists a $B$ with a retraction $r : \Omega \to B$? (That's all of them.) Are you asking for spaces such that for all $B \subset \Omega$, there exists a retraction $\Omega \to B$? (Very few spaces.) Does subspace mean proper subspace, or any subspace? In any case I'm not sure that your question is research-level. $\endgroup$– Najib IdrissiCommented Nov 23, 2018 at 15:28
-
1$\begingroup$ I will assume that you mean: given $B \subset \Omega$, how to see if there exists a retraction $\Omega \to B$? (There is a reason quantifiers are usually put at the beginning...) Anyway, this is too broad. You have some tools from algebraic topology but otherwise, I doubt you can get a better answer than "it exists when it exists". $\endgroup$– Najib IdrissiCommented Nov 23, 2018 at 15:49
-
3$\begingroup$ $\mapsto$ means assignment, the functional notation is $\to$ $\endgroup$– YCorCommented Nov 23, 2018 at 17:08
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
If $\Omega$ is a nonempty space and $x\in \Omega$, then the constant map sending everything to $x$ is a retraction from $\Omega$ to $\{x\}$. So, every nonempty topological space has retracts.
answered Nov 23, 2018 at 15:10
-
2$\begingroup$ Just take $B = \Omega$ and $r = \operatorname{id}$. Besides the empty space retracts to itself. (If you reply that you only consider proper subspaces, then you need to remove singletons from your answer.) (Also, two upvotes for an answer to an undergrad-level question?) $\endgroup$ Commented Nov 23, 2018 at 15:26
-
$\begingroup$ @NajibIdrissi : I miss-formulated (so I will edit my question). Two downvotes, by the way ... $\endgroup$– THCCommented Nov 23, 2018 at 15:40
-
$\begingroup$ @THC Your question got three downvotes and one upvote, this answer got two upvotes for some reason. $\endgroup$ Commented Nov 23, 2018 at 15:41