Structure on union of topological spaces

Calculus Level 3

True or False: If $\ell$ is a limit point of $\bigcup_\alpha U_\alpha$ , then $\ell$ is also a limit point of at least one of the $U_\alpha$ .

False True

2 solutions

Hobart Pao
Mar 31, 2019

Counter example: take in $\mathbb{R}$ all intervals of the form $\left[ 0, 1-\dfrac{1}{n}\right] , n \in \mathbb{N}$ . Then each such interval has all its limit points and is closed, but the union of all such intervals is [0,1), which has the limit point 1 which is not found in any of those intervals.

Arghyadeep Chatterjee
Sep 1, 2019

Arbitrary union of closed sets is not closed . (Arbitrary intersection is closed) . Closed set means that each of it's point is an accumulation point of the set . So the result in the question follows

Structure on union of topological spaces

2 solutions

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