Exercise · ZW-PATH-CONNECTED-SPACE-002 · ★☆☆ Basic
Which is a path in \(\mathbb R\) from \(2\) to \(5\)?
Choose one option.
Show answer
- A. \(\gamma(t)=2+3t\)
Explanation
We need \(\gamma(0)=2\) and \(\gamma(1)=5\). Only \(2+3t\) satisfies both; \(5-3t\) goes from \(5\) to \(2\), \(3t\) starts at \(0\) and \(2+5t\) ends at \(7\).
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
- ZW-PATH-CONNECTED-SPACE-001 Which of these subsets of mathbb R is path-connected?
- ZW-PATH-CONNECTED-SPACE-003 Is the space X=\1,2,3\ with the indiscrete topology path-connected?
- ZW-PATH-CONNECTED-SPACE-004 Why is \1,2\ with the discrete topology not path-connected?
- ZW-PATH-CONNECTED-SPACE-005 One of these topologies on X=\1,2,3\ does not make X path-connected. Which one?
- ZW-PATH-CONNECTED-SPACE-006 Which statement is true?
- ZW-PATH-CONNECTED-SPACE-007 Which of these subsets of mathbb R^2 is path-connected?
- ZW-PATH-CONNECTED-SPACE-008 If α joins a to b and β joins b to c, which path joins a to c?
- ZW-PATH-CONNECTED-SPACE-009 How many of the 29 topologies on a 3-point set make it path-connected?