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Topology: An Inquiry-Based Approach

Steven Schlicker

Section Bases for Subspaces

Recall that a basis \(\B\) for a topological space is a collection of sets that generate all of the open sets through unions. If we have a basis \(\B\) for a topological space \((X, \tau)\text{,}\) and if \(A\) is a subspace of \(X\text{,}\) we might ask if we can find a basis \(\B_A\) from \(\B\) in a natural way.

Activity 15.3.

Let \((X, \tau)\) be a topological space with basis \(\B\text{,}\) and let \(A\) be a subspace of \(X\text{.}\)

(a)

There is a natural candidate to consider as a basis \(\B_A\) for \(A\text{.}\) How do you think we should define the elements in \(\B_A\text{?}\)

(b)

Recall that a set \(\B\) is a basis for a topological space \(X\) if
  1. For each \(x \in X\text{,}\) there is a set in \(\B\) that contains \(x\text{.}\)
  2. 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 \(x \in B_3 \subseteq B_1 \cap B_2\text{.}\)
Show that your set from (a) is a basis for the induced topology on \(A\text{.}\)
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