Section 16: The Subspace Topology

• Equivalently, the subspace topology is generated by $\{A\cap U|U\mbox{ is a basis element of }\mathcal{T}\}$ .

Properties

• If $A$ is a subspace of $X$ , and $B$ is a subset of $A$ , then the subspace topologies $B\subset A$ and $B\subset X$ agree.
• If $A$ is open in $X$ , and $B$ is open in $A$ , then $B$ is open in $X$ .
• The product topology on $A\times B$ is the same as the subspace topology on $A\times B\subset X\times Y$ .
• If $X$ is ordered, the order topology on $A$ is, in general, not the same as the subspace topology on $A\subset X$ (but it is always coarser).
  • As an example, consider $X=[0,1]\times[0,1]$ with the product topology, $X_{dic}$ with the dictionary order topology (the ordered square, $I_{o}^{2}$ ), and $X_{subdic}$ with the subspace topology inherited from $\mathbb{R}^{2}$ in the dictionary order topology (the latter is the same as the product topology $\mathbb{R}_{d}\times\mathbb{R}$ ). Then $X_{subdic}$ is strictly finer than $X_{dic}$ and $X$ , where the latter two topologies are not comparable.

• If $X$ is ordered and $A$ is convex, then the order topology on $A$ agrees with the subspace topology on $A\subset X$ .