Exercise 3, § 2

Let $X, X'$ be two topological spaces, $f$ a bijection of $X$ onto $X'$: then a necessary and sufficient condition for $f$ to be a homeomorphism

(*) For an example of a space which satisfies $(D_{IV})$ but neither $(D_{II})$ nor $(D_{III})$, see $§ 8$, Exercise 6 b). For an example of a space satisfying $(D_{II})$ and $(D_{III})$, but not $(D_I)$, see Chapter IX, $§ 5$, Exercise 16. For examples of spaces where one of the two conditions $(D_{II}), (D_{III})$ is satisfied but not the other, see $§ 9$, Exercise 23.

\* In a metrizable space, $(D_I)$ and $(D_{IV})$ are equivalent: Proposition 12 of Chapter IX, $§ 2$, no. 8 shows that $(D_I)$ and $(D_{II})$ are equivalent; on the other hand, if $(D_{IV})$ is satisfied, then for each integer $n > 0$ there is a maximal countable family $\mathfrak{B}_n = (B_{nm})_{m \geq 0}$ of mutually disjoint balls of radius $1/n$; whose union is dense. The countable set of centres of the $B_{nm}$ is then dense. \* of $X$ onto $X'$ is that the topology of $X'$ is the finest for which $f$ is continuous.