Exercise · ZW-CONNECTED-COMPONENT-010 · ★★★ Advanced
A space has finitely many connected components. Which statement is true?
Choose one option.
Show answer
- C. Each component is both open and closed.
Explanation
Components are always closed; the complement of one is a finite union of closed sets, hence closed, so the component is also open.
Review the concept: Connected component
The connected component \(C(x)\) of a point \(x\) of a space \(X\) is the largest connected subset \(A\subseteq X\) containing \(x\).
\[C(x)=\bigcup\{A\subseteq X : x\in A,\ A\text{ connected}\}\]
More practice problems: connected component
- ZW-CONNECTED-COMPONENT-001 In X=\1,2,3,4\ the open sets are the unions of \1,2\,\3\,\4\. What is the connected component of 1?
- ZW-CONNECTED-COMPONENT-002 How many connected components does ℝsetminus\0,1\ have with the usual topology?
- ZW-CONNECTED-COMPONENT-003 How many connected components does a discrete space with 5 points have?
- ZW-CONNECTED-COMPONENT-004 In ℚ with the usual topology, what is the connected component of 0?
- ZW-CONNECTED-COMPONENT-005 Which statement about the connected components of a space is true?
- ZW-CONNECTED-COMPONENT-006 In X=\1,2,3\ with open sets ∅,\1\,\2,3\,X, what is the connected component of 2?
- ZW-CONNECTED-COMPONENT-007 Among the 29 topologies on \1,2,3\, how many have exactly 3 connected components?
- ZW-CONNECTED-COMPONENT-008 Among the 29 topologies on \1,2,3\, how many have exactly 2 connected components?