Let $f:X \rightarrow Y$ be a map of topological spaces. There are several characterizations for $f$ to be continuous, closed, or open, usually in terms of pre-images, closures, and interiors of sets. Here is what I think I know:

$f$ is continuous $\iff$ $\forall B \subseteq Y, \overline{f^{-1}(B)}\subseteq f^{-1}(\overline{B})$ $\iff$ $\forall A\subseteq X, f(\overline{A})\subseteq \overline{f(A)}$.

$f$ is closed $\iff$ $\forall A \subseteq X$, $\overline{f(A)} \subseteq f(\overline{A})$

$f$ is open $\iff$ $\forall A \subseteq X$, $f(A^\circ) \subseteq f(A)^\circ$ $\Rightarrow \forall B\subseteq Y, f^{-1}(\overline{B})\subseteq \overline{f^{-1}(B)}$. (I'm actually not sure if the reverse implication holds for this last statement).

Is there some "comprehensive" list of characterizations similar to these? Does the reverse implication hold for the last statement?


1 Answer 1


I would say that $f$ continuous iff every inverse image of an open set is open is the default definition. All other statements equivalent to it are "characterisations" of continuity in terms of different notions. They can serve as definitions in other contexts; e.g. when we define a closure space instead of a topology, we can use the characterisation using closures from topology as a motivation for a definition of "closure-continuity", as it were.

Ditto for $f$ open iff images of open sets are open, and closed maps too. The other equivalences I would not call definitions but characterisations of that notion. Similarly we could then define "closure-closed map" in a closure space, etc.

For continuity we can add

$$\forall B \subseteq Y: f^{-1}[B^\circ] \subseteq (f^{-1}[B])^\circ\tag{1}$$ $$\forall B \subseteq Y: \partial f^{-1}[B] \subseteq f^{-1}[\partial B]\tag{2}$$ to the list. The last one in terms of the boundary $\partial$ of sets (which is of rarer interest).

Characterisations of open and closed maps are rarer (in my books at least).

I believe the following reverse inclusions (i.e. reverse to the continuity case) do characterise openness of $f$:

$$\forall B \subseteq Y: f^{-1}[\partial B] \subseteq \partial f^{-1}[B]\tag{3}$$ $$\forall B \subseteq Y: f^{-1}[\overline{B}] \subseteq \overline{f^{-1}[B]}\tag{4}$$

(I have some proofs in this answer; see also this question etc.)

I haven't seen another one in terms of interior, or a closed map characterisation in interior-terms. Maybe look for those yourself.

  • $\begingroup$ Thanks. You're right, "characterizations" is more appropriate (I have edited the question). A proof of the reverse implication I asked for can be found here: math.stackexchange.com/questions/393354/… $\endgroup$
    – Ben
    Feb 8, 2020 at 13:46

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