Relative topology

2020 Mathematics Subject Classification: Primary: 54B05 [MSN][ZBL]

of a subset $A$ of a topological space $(X,\tau)$

The system $\tau_A$ of intersections of all possible open subsets of $(X,\tau)$ (i.e. of elements of the topology $\tau$) with $A$: $$\tau_A = \{ A \cap U : U \in \tau \} \ .$$ The relative topology is often called the induced topology or subspace topology.

A subset of the topological space $(X,\tau)$ equipped with the relative topology is called a subspace of $(X,\tau)$. A subspace of a $T_i$-space is itself a $T_i$-space, $i=0,1,2,3,3\frac{1}{2}$ (cf. Separation axiom). A subspace of a metrizable space is itself metrizable. Any Tikhonov space of weight $\leq\theta$ (that is, having an open base of cardinality $\leq \theta$) is homeomorphic to a subspace of a Hausdorff compactum of weight $\leq\theta$ by Tikhonov's theorem.

Comments

The subspace topology is the coarsest topology on $A$ for which the embedding map $A \hookrightarrow X$ is continuous.

Topological properties which pass to subspaces are called hereditary.

References

[a1]	J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 50ff