Question

Show that every finite set is a closed set.
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Answer 1

Answer:

If you take with the standard topology any finite set is closed as it is the complement of an open set. The open intervals form a basis for the standard topology. The complement of a finite set is precisely the union of open sets.

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Answer 2

Each finite set is closed set

Explanation:

• Thinking A as a subset of metric space M.
• An easy approach will be to use that the single points ${X_{j}}\subset M$ are closed, then of course$A=Union\hspace{0.2 cm}of\hspace{0.2 cm} X_{j}$

• from j=1 to j=n.

• Since A is a finite union of closed set, it is itself closed.

Also, If we show that the complement of A is open, then A is closed.

• Let $B=A^{c}$, and suppose that y∈B. We need to show that there is an r>0 for which Br(y)∩A=∅, that is Br(y)⊂B.

• If we let r=min{d(y,xi) : i=1,...,n} then it must be true that Br(y)∩A=∅, since for every z∈Br(y) we have d(z,y)<d(y,xi) for each i=1,...,n.

• This clearly shows that the complement of A is open which means A should be closed.

Hence, Every finite set is a closed set.