Exercise · ZW-CONNECTED-SPACE-007 · ★★☆ Intermediate

In \(X=\{1,2,3,4\}\) with open sets \(\varnothing, \{1\}, \{2\}, \{1,2\}, \{3,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}\), in how many ways can \(X\) be split into two disjoint nonempty open sets (order ignored)?

Choose one option.

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  • B. 3

Explanation

We look for pairs \(\{U,V\}\) of nonempty open sets with \(U\cap V=\varnothing\) and \(U\cup V=X\): 3 pairs in total, counted by checking every pair of open sets.

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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

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