Relative topology
2020 Mathematics Subject Classification: Primary: 54B05 [MSN][ZBL]
of a subset 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 |
Relative topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_topology&oldid=55844