Topology/Subspaces

From Wikibooks, open books for an open world
Jump to: navigation, search
Topology
 ← Sequences Subspaces Order → 


Put simply, a subspace is analogous to a subset of a topological space. Subspaces have powerful applications in topology.

Definition[edit]

Let be a topological space, and let be a subset of . Define the open sets as follows:

A set is open in if there exists a a set such that

An important idea to note from the above definitions is that a set not being open or closed does not prevent it from being open or closed within a subspace. For example, as a subspace of itself is both open and closed.


Topology
 ← Sequences Subspaces Order →