Question

Let E be a subset of metric space X
Such that no limit point of E then,
Select one:
A. E is open
B. E is not open
O C. E is not closed.
O D. E is closed​

Answer 1

Answer:

I am not sure if you are not replying to this email

Answer 2

D. E is closed​

GIVEN: E is a subset of metric space X
TO PROOF: E is closed.
SOLUTION:

According to the question,

Let us assume that E is closed.

Now,

x∈ Ec, then x∉E.
So,

x is not the limit point of E.

Hence,

There is a neighborhood N of x.

Such that,

E∩N is empty, where  N⊂Ec.

Thus, x is an interior point of Ec and Ec is open.

Now,

Let us assume Ec is open.

Now, let x be a limit point of E, then for every neighborhood of x that contains a point of E such that x is not an interior point of Ec.

Since Ec is open,

This implies that x∈E,

Therefore E is closed.

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