Proposition 3

Let X, Y be two topological spaces, let $f : X \to Y$ be a continuous mapping, let R be the equivalence relation $f(x) = f(y)$ on X, and let $X \xrightarrow{p} X/R \xrightarrow{h} f(X) \xrightarrow{i} Y$ be the canonical decomposition of f. Then the following three statements are equivalent:

a) f is an open mapping.

b) The three mappings p, h, i are open.

c) The equivalence relation R is open, h is a homeomorphism, and $f(X)$ is an open subset of Y.

Also the preceding proposition remains true if "open" is replaced by "closed" throughout.

Since the injection i is continuous, it follows from Proposition 1 c) of no. 1 that if f is open then so is $h \circ p$; since p is surjective and continuous, Proposition 1 b) shows that h is open; h is in any case a continuous bijection; thus h is a homeomorphism, and therefore, from Proposition 1 a) of no. 1, $p = \overline{h}^{-1} \circ (h \circ p)$ is an open mapping. Also [no. 1, Proposition 1 b)] $i \circ h$ is open; hence [no. 1, Proposition 1 a)] so is $i = (i \circ h) \circ \overline{h}^{-1}$. This proves that a) implies b). Conversely, Proposition 1 a) of no. 1 shows that b) implies a). Finally, the equivalence of b) and c) follows immediately from the definitions.

The proof in the case of closed mappings is analogous, mutatis mutandis.