Exercise · ZW-CONNECTED-SPACE-010 · ★★★ Advanced
Let \(A,B\subseteq X\) be connected (as subspaces) with \(A\cap B\neq\varnothing\). Then \(A\cup B\) is:
Choose one option.
Show answer
- B. connected
Explanation
If \(A\cup B\) split into disjoint open sets, the common point would lie in one of them and each connected set \(A\), \(B\) would lie entirely in the one containing that point. So there is no separation.
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?