Proposition 5

Let $X, Y$ be two topological spaces, $f$ a mapping of $X$ into $Y, \mathfrak{B}$ a base of the topology of $X$. Then the following statements are equivalent:

a) $f$ is an open mapping.

b) For each $U \in \mathfrak{B}$, $f(U)$ is open in $Y$.

c) For each $x \in X$ and each neighbourhood $V$ of $x$ in $X$, $f(V)$ is a neighbourhood of $f(x)$ in $Y$.

The equivalence of a) and b) follows immediately from the definitions and from $(O_1)$; the equivalence of a) and c) is a consequence of Proposition 1 of § 1, no. 2.