Exercise · ZW-CONTINUOUS-MAP-TOPOLOGICAL-003 · ★☆☆ Basic
What must a function \(f:X\to Y\) between topological spaces satisfy to be continuous?
Choose one option.
Show answer
- D. The preimage of each open set of \(Y\) is open in \(X\)
Explanation
Continuity is a condition on preimages of open sets. Requiring images of open sets to be open defines open maps, and requiring preimages of points to be open is much stronger.
Review the concept: Continuous function between topological spaces
A function \(f:X\to Y\) between topological spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) is continuous if the preimage of every open set \(V\) of \(Y\) is an open set of \(X\).
\[V\in\mathcal{T}_Y\ \Longrightarrow\ f^{-1}(V)\in\mathcal{T}_X\]
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