Example 1

Inverse image of a topology. Let $X$ be a set, $Y$ a topological space, $f$ a mapping of $X$ into $Y$; the coarsest topology $\mathfrak{T}$ on $X$ for which $f$ is continuous is called the inverse image under $f$ of the topology of $Y$. It follows from Proposition 4 and the formulae for the inverse image of a union and an intersection [Set Theory, R, § 4, formulae (34) and (46)] that the open (resp. closed) sets in the topology $\mathfrak{T}$ are the inverse images under $f$ of the open (resp. closed) sets of $Y$; consequently, for each $x \in X$, the sets $f^{-1}(W)$, where $W$ runs through a fundamental system of neighbourhoods of $f(x)$ in $Y$, form a fundamental system of neighbourhoods of $x$ in the topology $\mathfrak{T}$. In § 3 we shall study, under the name of induced topology, the particular case in which $X$ is a subset of $Y$ and $f$ is the canonical injection $X \to Y$; $X$, with the induced topology, is then called a subspace of $Y$.

For a mapping $f$ of a topological space $X$ into a topological space $X'$ to be continuous it is necessary and sufficient that the topology of $X$ is finer than the inverse image under $f$ of the topology of $X'$.