Exercise · ZW-PATH-CONNECTED-SPACE-010 · ★★★ Advanced

Let \(X=\{1,2,3\}\) with open sets \(\varnothing,\{1\},\{3\},\{1,3\},X\). Is it path-connected?

Choose one option.

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  • C. Yes: for example \(\gamma(t)=1\) if \(t<1\) and \(\gamma(1)=2\) joins \(1\) to \(2\)

Explanation

The only open set containing \(2\) is \(X\), so preimages of open sets are \([0,1)\), \(\varnothing\) or \([0,1]\), all open. Likewise \(3\) is joined to \(2\).

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Review the concept: Path-connected space

A space \(X\) is path-connected if every pair of points \(a,b\in X\) is joined by a path: a continuous function \(\gamma:[0,1]\to X\) with \(\gamma(0)=a\) and \(\gamma(1)=b\).

\[\forall a,b\in X\ \exists\,\gamma:[0,1]\to X\text{ continuous},\ \gamma(0)=a,\ \gamma(1)=b\]

More practice problems: path-connected space

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