Example

The topology induced on the set $\mathbf{Z}$ of rational integers by the topology of the rational line is the discrete topology, for the intersection of $\mathbf{Z}$ and the open interval $]n - 1/2, n + 1/2[$ is the set $\{ n \}$.

By Proposition 5 of § 2, no. 3 (or directly from Definition 1) we see that, if $B \subset A \subset X$, the subspace $B$ of $X$ is identical with the subspace $B$ of the subspace $A$ of $X$ (transitivity of induced topologies). If $\mathfrak{G}$ is a subbase (resp. a base) of the topology of $X$ ($§ 2$, no. 3, Example 3) its trace $\mathfrak{G}_A$ on $A$ is a subbase (resp. a base) of the topology induced on $A$.

In all questions which involve the elements or subsets of $A$, it is essential to distinguish carefully between their properties as points (resp. subsets) of $X$, and their properties as points (resp. subsets) of the subspace $A$. We shall make this distinction by using the phrases "in $A$", "with respect to $A$", or "relative to $A$" to refer to properties in the latter category (possibly contrasting them with the phrases "in $X$", "with respect to $X$", "relative to $X$").

An open set of the subspace $A$ need not be open in $X$; in order that every set which is open in $A$ should be open in $X$ it is necessary and sufficient that $A$ is open in $X$. The condition is necessary, since $A$ is open in $A$, and it is sufficient by virtue of $(O_{II})$ and Definition 1.

The sets which are closed in $A$ are the intersections with $A$ of the closed sets in $X$ ($§ 2$, no. 3, Example 1); as above we see that every set which is closed in $A$ is closed in $X$ if and only if $A$ is closed in $X$.

The neighbourhoods of a point $x \in A$ relative to $A$ are the intersections with $A$ of neighbourhoods of $x$ relative to $X$. Every neighbourhood of $x$ relative to $A$ is a neighbourhood of $x$ relative to $X$ if and only if $A$ is a neighbourhood of $x$ in $X$.