Subspace topology

The open sets of \(A\) are obtained by trimming those of \(X\): each open set \(U\) of \(X\) contributes the set \(A\cap U\). This family satisfies the axioms of a topology because \(A\cap\bigcup U_i=\bigcup(A\cap U_i)\) and \(A\cap(U\cap W)=(A\cap U)\cap(A\cap W)\). Different open sets of \(X\) can give the same open set of \(A\), so \(A\) often has fewer open sets than \(X\).

The closed sets of \(A\) are the intersections \(A\cap F\) with \(F\) closed in \(X\), and the inclusion \(i:A\to X\) is continuous because \(i^{-1}(U)=A\cap U\). In fact it is the smallest topology on \(A\) that makes the inclusion continuous. It is also transitive: if \(A\subseteq B\subseteq X\), the topology \(A\) inherits from \(B\) is the one it inherits from \(X\).

A set being open in the subspace does not mean it is open in the larger space. In \(A=[0,1]\subseteq\mathbb{R}\), the set \([0,\tfrac12)=(-1,\tfrac12)\cap A\) is open in \(A\) but not in \(\mathbb{R}\). Only when \(A\) itself is open in \(X\) are the open sets of \(A\) open in \(X\). Likewise \(\mathbb{Z}\subseteq\mathbb{R}\) inherits the discrete topology, since \(\{n\}=(n-\tfrac12,n+\tfrac12)\cap\mathbb{Z}\).

This construction allows us to talk about continuity of functions defined on parts of a space, such as a curve on an interval, and it underlies the product topology and hereditary properties. It is the step before studying compact and connected subspaces.