Exercise · ZW-CONTINUOUS-MAP-TOPOLOGICAL-006 · ★★☆ Intermediate
Let \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=x^2\), with the usual topology. What is \(f^{-1}\big((1,4)\big)\)?
Choose one option.
Show answer
- C. \((-2,-1)\cup(1,2)\)
Explanation
We need \(1<x^2<4\), i.e. \(1<|x|<2\): \(x\in(-2,-1)\cup(1,2)\). It is open in \(\mathbb{R}\), as continuity requires.
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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