Exercise · ZW-CONTINUOUS-MAP-TOPOLOGICAL-004 · ★☆☆ Basic
Let \(X\) be a set with at least two points. Which statement is true?
Choose one option.
A The identity from \(X\) with the trivial topology to \(X\) with the discrete one is continuous B The identity from \(X\) with the discrete topology to \(X\) with the trivial one is continuous C No constant function between topological spaces is continuous D The identity from usual \(\mathbb{R}\) to discrete \(\mathbb{R}\) is continuous Show answer B. The identity from \(X\) with the discrete topology to \(X\) with the trivial one is continuous← Back to the question
Explanation The open sets of the trivial topology are \(\varnothing\) and \(X\), whose preimages are \(\varnothing\) and \(X\), open in the discrete one. In the other direction, \(\{x\}\) is open in the target but not in the domain . Constants are continuous, since every preimage is \(\varnothing\) or the whole domain.
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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