Are bijective functions in stone spaces continuous? I get that bijectivity doesn't imply continuity in general, but is this different in stone spaces?
Are bijective functions in stone spaces continuous?
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general-topology
functions
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0stone spaces $\to$ Stone spaces. In this way, the question is less hard. – 2017-01-17
1 Answers
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No, they are not in general. Stone spaces have a lot of structure, despite being totally disconnected.
One easy counterexample is the following. Consider the Cantor space $X=2^\omega$ of infinite binary sequences. This is clearly a Stone space. Now let $f$ be the identity on $X$, except that it swaps the all-$0$s and all-$1$s sequences.
This $f$ is clearly a bijection, but it's not hard to show that it is not continuous. Specifically, note that $f(\overline{0})$ is in the open set $V=\{$sequences beginning with $1\}$, but it has no neighborhood $U\ni \overline{0}$ which is mapped into $V$.
Indeed, this can be generalized. Let $Y$ be an arbitrary Stone space; for $a,b\in Y$, let $f_{a, b}$ be the map which is the identity on $Y\setminus\{a, b\}$, and which swaps $a$ and $b$. This $f$ is a bijection. When is it continuous?
Well, let $U, V$ be open sets with $a\in U, b\in V$, and $U\cap V=\emptyset$ (these exist since $Y$ is Hausdorff). Then the $f$-preimage of $V$ is $W=(V\setminus\{b\})\cup\{a\}$; in order for $f$ to be open, $W$ must be open too. But then $W\cap U$, being the intersection of two open sets, is open; and $W\cap U=\{a\}$.
Thinking about this argument, we get a general fact:
If $X$ is a Hausdorff space such that every self-bijection is continuous, then $X$ is discrete.
In particular, the compactness part of Stone spaces doesn't enter into this.
Incidentally, Hausdorffness is necessary: consider the cofinite topology on an infinite set. This topology is $T_1$ and compact, and every self-bijection is continuous; yet it is not discrete.