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Section Summary
Important ideas that we discussed in this section include the following.
A topology on a set \(X\) is a collection of open subsets of \(X\text{.}\) More specifically, a set \(\tau\) of subsets of a set \(X\) is a topology on \(X\) if
\(X\) and \(\emptyset\) belong to \(\tau\text{,}\)
any union of sets in \(\tau\) is a set in \(\tau\text{,}\) and
any finite intersection of sets in \(\tau\) is a set in \(\tau\text{.}\)
A topological space is a set along with a topology on the set.
Any arbitrary union of open sets is open and any finite intersection of open sets is open in a topological space.
It can be difficult to completely describe the open sets in a topology, and it can be difficult to work with arbitrary open sets. If a collection of simpler sets generate a topology, that collection of simpler sets is a basis for the topology. More formally set \(\B\) is a basis for a topology on a set \(X\) if
For each \(x \in X\text{,}\) there is a set in \(\B\) that contains \(x\text{.}\)
If \(x \in X\) is an element of \(B_1 \cap B_2\) for some \(B_1, B_2 \in B\text{,}\) then there is a set \(B_3 \in \B\) such that \(x \in B_3 \subseteq B_1 \cap B_2\text{.}\)
A subset \(A\) of a topological space \(X\) is a neighborhood of a point \(a \in A\) if there is an open set \(O\) contained in \(A\) such that \(a \in O\text{.}\)
A point \(x\) in a subset \(A\) of a topological space \(X\) is an interior point of \(A\) if \(A\) is a neighborhood of \(x\text{.}\) The interior of set \(A\) is the collection of all interior points of \(A\text{.}\)
A subset \(A\) of a topological space \(X\) is open if and only if \(A\) is equal to its interior.
