Lemma 5.6.5. Let f : X \to Y be surjective, closed, continuous map of topological spaces. Let T \subset Y be a subset. Then
\overline{T} = f(\overline{f^{-1}(T)}),
T \subset Y is closed if and only if f^{-1}(T) is closed,
T \subset Y is open if and only if f^{-1}(T) is open, and
T \subset Y is locally closed if and only if f^{-1}(T) is locally closed.
In particular we see that f is submersive.
Proof.
It is clear that \overline{f^{-1}(T)} \subset f^{-1}(\overline{T}). Then T \subset f(\overline{f^{-1}(T)}) \subset \overline{T} is a closed subset, hence we get (1). Part (2) is obvious from the fact that f is closed and surjective. Part (3) follows from (2) applied to the complement of T. For (4), if f^{-1}(T) is locally closed, then f^{-1}(T) \subset \overline{f^{-1}(T)} is open. Since the map \overline{f^{-1}(T)} \to \overline{T} is surjective by (1) we can apply part (3) to the map \overline{f^{-1}(T)} \to \overline{T} induced by f to conclude that T is open in \overline{T}, i.e., T is locally closed.
\square
Comments (2)
Comment #3562 by Laurent Moret-Bailly on
Comment #3686 by Johan on
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