O'Reilly logo

Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

4.9 Subspaces

A subspace Y of a topological space X is just a subset of X. However, one needs to impose a topology on Y to make it a subspace.

Definition 4.9.1 (Subspace topology) Let X be a topological space, and Y be a subset of X. The subspace topology on Y has as opens the subsets of Y of the form UY, U open in X.

One then says that Y, with the subspace topology, is a subspace of X.

There is a slightly more general definition.

Definition 4.9.2 (Induced topology) Given any map f : ZX, where Z is a set and X is a topological space, the topology on Z induced from that of X by f is the coarsest topology on Z that makes f continuous.

The induced topology is simple: take as opens in Z exactly those subsets of the form f−1(U), U open in

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required