Let X=\1,2,3\ have open sets ∅,\1\,\1,2\,X and q:X→\a,b\ with q(1)=a, q(2)=q(3)=b. How many open sets does the quotient topology have?

Exercise · ZW-QUOTIENT-TOPOLOGY-008 · ★★☆ Intermediate

Let \(X=\{1,2,3\}\) have open sets \(\varnothing,\{1\},\{1,2\},X\) and \(q:X\to\{a,b\}\) with \(q(1)=a\), \(q(2)=q(3)=b\). How many open sets does the quotient topology have?

Choose one option.

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B. \(3\)

Explanation

The preimages are \(q^{-1}(\{a\})=\{1\}\) (open) and \(q^{-1}(\{b\})=\{2,3\}\) (not open). The open sets are \(\varnothing\), \(\{a\}\) and \(\{a,b\}\): 3.

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Review the concept: Quotient topology

The topology \(\mathcal{T}_q\) on \(Y\) determined by a surjective function \(q:X\to Y\) from a space \((X,\mathcal{T})\), whose open sets are the \(V\subseteq Y\) with \(q^{-1}(V)\) open in \(X\).

\[\mathcal{T}_q=\{V\subseteq Y:\ q^{-1}(V)\in\mathcal{T}\}\]

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