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Show by counterexample that a connected component of a topological space is not necessarily open.

Show by counterexample that a connected component of a topological space is not necessarily open.

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Answer #1

ANSWER :-

Here, we need to show that a connected component of a topological space is not necessarily open.

Example :

if the set is {1/2,2/2,3/2,.....,0} is completely disconnected component.

because the elements are not singletons they are in the set form.

if the set is {1/2},{2/2},{3/2},.....,{0} is completely connected component.

meets to zero, so is inevitably in any open set around 0. Along these lines {0} neglects to be open. To demonstrate {0} is a segment see that for every n

so, a connected component of a topological space is need not necessarily open.

Thank you

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