Discussion
In a topological space, which of the following statements is NOT always true :
Answer: B
The statement that is not always true is Union of any family of closed sets is closed.
Only finite unions of closed sets are guaranteed closed. In \(\mathbb{R}\), the closed sets \([1/n, 1]\) for \(n = 1, 2, \ldots\) have union \((0, 1]\), which is not closed.
The other three always hold. A finite union of compact sets is compact, since a finite collection of finite subcovers is still finite. Connected sets sharing a common point have a connected union, because any separation would have to split one of the sets. A union of dense subsets contains a dense subset, and any superset of a dense set is dense.
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