Exercise · ZW-QUOTIENT-TOPOLOGY-003 · ★☆☆ Basic
Let \(q:X\to Y\) be surjective and \(X\) a topological space. When is a set \(V\subseteq Y\) open in the quotient topology?
Choose one option.
Show answer
- A. When \(q^{-1}(V)\) is open in \(X\)
Explanation
That is the definition: the topology is pulled back through preimages. Requiring images to be open is a different condition (an open map) and does not define the quotient topology.
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 \(X\).
\[\mathcal{T}_q=\{V\subseteq Y:\ q^{-1}(V)\in\mathcal{T}\}\]
More practice problems: quotient topology
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