Exercise · ZW-CONNECTED-SPACE-009 · ★★★ Advanced
A space \(X\) is connected if and only if the only subsets of \(X\) that are both open and closed are:
Choose one option.
Show answer
- C. \(\varnothing\) and \(X\)
Explanation
If \(U\) is clopen, \(U\) and \(X\setminus U\) are disjoint open sets with union \(X\); this is a separation unless \(U=\varnothing\) or \(U=X\). Conversely, a separation \(U,V\) makes \(U\) a proper clopen set.
Review the concept: Connected space
A topological space \(X\) is connected if it cannot be written as the union of two disjoint nonempty open sets \(U\) and \(V\).
\[X=U\cup V,\ U\cap V=\varnothing\ \Rightarrow\ U=\varnothing\ \text{ or }\ V=\varnothing\]
More practice problems: connected space
- ZW-CONNECTED-SPACE-001 Which of these topologies on X=\1,2,3\ is connected?
- ZW-CONNECTED-SPACE-002 How many of the 29 topologies on a 3-point set are connected?
- ZW-CONNECTED-SPACE-003 In the usual ℝ, which of these subsets is connected?
- ZW-CONNECTED-SPACE-004 The subspace ℝsetminus\0\ of ℝ is not connected. Which pair of open sets separates it?
- ZW-CONNECTED-SPACE-005 A discrete space with n≥ 1 points is connected exactly when:
- ZW-CONNECTED-SPACE-006 If X is connected and f:X→ℝ is continuous, what is f(X) always?
- ZW-CONNECTED-SPACE-007 In X=\1,2,3,4\ with open sets ∅, \1\, \2\, \1,2\, \3,4\, \1,3,4\, \2,3,4\, \1,2,3,4\, in how many w…
- ZW-CONNECTED-SPACE-008 The open sets ℚ∩(-∞,√2) and ℚ∩(√2,∞) separate ℚ. Which of these pairs does not?