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Section Summary
Important ideas that we discussed in this section include the following.
A subspace of a topological space is any nonempty subset of the topological space endowed with the subspace topology.
An open subset in the subspace topology for a subset \(A\) of a topological space \(X\) is any set of the form \(O \cap A\text{,}\) where \(O\) is an open set in \(X\text{.}\)
The relatively open sets are the open sets in a subspace topology. The relatively closed sets are complements of the relatively open sets in a subspace topology. That is, a relatively closed set in the subspace \(A\) of a topological space \(X\) are the sets of the form \(A \cap C\text{,}\) where \(C\) is a closed set in \(X\text{.}\)
The topological space \(\R\) with the standard topology is homeomorphic to any open interval as well as open intervals of the form \((a,\infty)\) or \((-\infty,b)\) for any real numbers \(a\) and \(b\text{.}\)
