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.
[edit] Definition
Let 
be a topological space, and let X1 be a subset of X. Define the open sets as follows:
A set 
is open in X1 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, (0,1) as a subspace of itself is both open and closed.
