Exercise · ZW-PATH-CONNECTED-SPACE-004 · ★★☆ Intermediate
Why is \(\{1,2\}\) with the discrete topology not path-connected?
Choose one option.
Show answer
- D. Because a path would be constant: its image is connected
Explanation
A path \(\gamma\) from \(1\) to \(2\) would have a connected image in a discrete space, hence a single point, and could not go from \(1\) to \(2\). Constant maps are continuous.
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-002 Which is a path in mathbb R from 2 to 5?
- ZW-PATH-CONNECTED-SPACE-003 Is the space X=\1,2,3\ with the indiscrete topology 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?