What is an example of a connected $T_2$-space $(X,\tau)$ with $X$ infinite and the following property?
If $\alpha \leq |X|$ is a non-empty cardinal, then there is a continuous map $f:X\to X$ such that $|f^{-1}(\{y\})| = \alpha$ for all $y\in X$.
$\begingroup$
$\endgroup$
4
asked Jan 8, 2020 at 15:58
-
3$\begingroup$ If the continuum hypothesis CH is true, then the circle is such a space. If CH is not true, then the circle is not such a space (a closed subset of the circle is either countable or has cardinality the continuum, any cardinal in between cannot be realized as the preimage of a point). $\endgroup$– Mathieu BaillifCommented Jan 9, 2020 at 0:16
-
$\begingroup$ Wow that's amazing @MathieuBaillif - thanks! $\endgroup$– Dominic van der ZypenCommented Jan 9, 2020 at 6:54
-
3$\begingroup$ What is the proof for a circle when alpha is infinite? $\endgroup$– TriCommented Jan 9, 2020 at 7:10
-
2$\begingroup$ For preimages of continuum cardinality, take the projection of a space (cylinder, rather) filling curve. For countable preimages, a function as in the answer to this question: math.stackexchange.com/questions/1599921/… $\endgroup$– Mathieu BaillifCommented Jan 9, 2020 at 9:04
Add a comment
|