Example Of A Non-Existent Retract

Question

I am looking for an example that disproves the claim that given any subspace $A$ of a topological space $X$, there exists a retract of $X$ onto $A$.

Answer 1

Here's the simplest possible example:

Consider the space $X$ with three points $a,b,c$ and open sets $$\emptyset, \{a\},\{c\}, \{a,c\}, \{a,b,c\}.$$ Let $A=\{a,c\}$. There are only two maps from $X$ to $A$ which are the identity on $A$, and neither is continuous. E.g. if we send $b$ to $a$, then the preimage of the open set $\{a\}$ is the non-open set $\{a,b\}$.

This really is the simplest example, since any space retracts onto any of its singleton subsets and onto itself.

Answer 2

A retract of a Hausdorff space is closed so $(0,1)$ is not a retract of $\mathbb{R}$ (usual topology).