Are there any topological spaces containing no proper retracts? i.e. $A\subset X$ s.t. $\exists r:X\to A$ continuous w/ $r(a)=a,\forall a\in A$

Question

Definition: Say that $A$ is a retract of a topological space $X$ if $A\subseteq X$ and there exists a continuous function (retraction) $r:X\to A$ such that $r(a)=a$ for all $a\in A$.

Is it the case that every topological space contains a proper retract?

I'm not sure if "proper retract" is a legitimate term, but by that I mean that $A$ is not just a single point, or the empty set, or $X$ itself.

It seems intuitive that any Euclidean space would have such a retract, but what about Hausdorff or metric spaces generally?

Are there any obvious counterexamples?

In your first paragraph, I think, you want to change $r:A\rightarrow X$. In addition to АрсенийКряжевiswithUkraine's counterexample one could also use the other boring counterexamples $\emptyset$ and ${●,○}$. — Severin Schraven, Aug 03 '22 at 02:26

score 7 · Accepted Answer · answered Aug 03 '22 at 07:05

The Cook continuum $C$ is a compact connected metrizable space with the surprising property that the only continuous functions $C\to C$ are the identity map and the constant functions.

Clearly no proper closed subspace containing at least two points of $C$ is a retract of $C$.

Are there any topological spaces containing no proper retracts? i.e. $A\subset X$ s.t. $\exists r:X\to A$ continuous w/ $r(a)=a,\forall a\in A$

1 Answers1