What is an example of a compact subset of a topological space X that is not closed
In topology, a subset of a topological space X is said to be compact if it is both closed and bounded. A closed set is one that contains all of its limit points, while a bounded set is one that can be contained within a ball of finite radius.
However, there are examples of compact subsets of topological spaces that are not closed. One such example is the Cantor set, which is a subset of the real numbers that is constructed by repeatedly removing the middle third of each interval in the set. The Cantor set is compact, as it is both closed and bounded. However, it is not closed, as it does not contain all of its limit points. For example, the number 1/2 is a limit point of the Cantor set, but it is not contained in the set itself.
Another example of a compact subset of a topological space that is not closed is the set of rational numbers between 0 and 1. This set is compact, as it is both closed and bounded. However, it is not closed, as it does not contain all of its limit points. For example, the number 1 is a limit point of the set, but it is not contained in the set itself.
Related Questions:
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- Is the union of two closed sets always closed?
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