Proposition 3

Given two topologies $\mathfrak{T}_1, \mathfrak{T}_2$ on a set $X$, the following statements are equivalent: a) $\mathfrak{T}_1$ is finer than $\mathfrak{T}_2$. b) For each $x \in X$, each neighbourhood of $x$ for $\mathfrak{T}_2$ is a neighbourhood of $x$ for $\mathfrak{T}_1$. c) For each subset $A$ of $X$, the closure of $A$ in the topology $\mathfrak{T}_1$ is contained in the closure of $A$ in the topology $\mathfrak{T}_2$. d) Every subset of $X$ which is closed in $\mathfrak{T}_2$ is closed in $\mathfrak{T}_1$. e) Every subset of $X$ which is open in $\mathfrak{T}_2$ is open in $\mathfrak{T}_1$.