Topology/Subspaces

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

DefinitionEdit

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
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