If α joins a to b and β joins b to c, which path joins a to c?

γ(t)=α(2t) if t≤tfrac12, β(2t-1) if t>tfrac12

γ(t)=α(t) if t≤tfrac12, β(t) if t>tfrac12

γ(t)=α(2t) if t≤tfrac12, β(2t) if t>tfrac12

Exercise · ZW-PATH-CONNECTED-SPACE-008 · ★★★ Advanced

If \(\alpha\) joins \(a\) to \(b\) and \(\beta\) joins \(b\) to \(c\), which path joins \(a\) to \(c\)?

Choose one option.

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B. \(\gamma(t)=\alpha(2t)\) if \(t\le\tfrac12\), \(\beta(2t-1)\) if \(t>\tfrac12\)

Explanation

Each piece must be reparametrized to \([0,1]\) and matched at \(t=\tfrac12\): \(\alpha(1)=b=\beta(0)\), so \(\gamma\) is continuous. Using \(\beta(2t)\) on the second piece leaves \([0,1]\).

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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 \(\gamma(1)=b\).

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

If \(\alpha\) joins \(a\) to \(b\) and \(\beta\) joins \(b\) to \(c\), which path joins \(a\) to \(c\)?

Explanation

Review the concept: Path-connected space

More practice problems: path-connected space