Topology-Subsequences
Let 
be a topological space, and let 
be a subset of 
. Define the open sets as follows:
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 
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.
as a subspace of itself is both open and closed.