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Dominic van der Zypen
Dominic van der Zypen
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Topological Infinite topological spaces such that every subset is a retract

Let $X$ be an infinite set and let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this imply that $\tau$ is either discrete ($\tau = \mathcal{P}(X)$) or indiscrete ($\tau = \{\emptyset, X\}$)?

Topological spaces such that every subset is a retract

Let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this imply that $\tau$ is either discrete ($\tau = \mathcal{P}(X)$) or indiscrete ($\tau = \{\emptyset, X\}$)?

Infinite topological spaces such that every subset is a retract

Let $X$ be an infinite set and let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this imply that $\tau$ is either discrete ($\tau = \mathcal{P}(X)$) or indiscrete ($\tau = \{\emptyset, X\}$)?

Post Undeleted by Dominic van der Zypen
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Source Link
Dominic van der Zypen
Dominic van der Zypen
  • 48.9k
  • 8
  • 47
  • 162

Topological spaces such that every subset is a retract

Let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this imply that $\tau$ is either discrete ($\tau = \mathcal{P}(X)$) or indiscrete ($\tau = \{\emptyset, X\}$)?