Answer to Question #116445 in Differential Geometry | Topology for Sheela John

Question #116445
Prove that the boundary of a subset A of a metric space X is always a closed set
1
Expert's answer
2020-05-20T19:55:58-0400

Let's fix some notation first.

"\\partial A" is the set denotes the boundary of "A" .

"M:=X\\backslash \\partial A= \\partial A^{c}" is the complement of the boundary of "A" .

We have given that "A\\subseteq X" ,where "X" is a metric space.


Claim1: No points in "M" can be the limit points of "\\partial A"

Proof: Suppose on the contrary there exist at least one point "q \\in M" is limit point of "\\partial A" .

Thus, for every "\\epsilon >0" there exist an open ball

"B(q,\\epsilon):=\\{y \\in X |\\: d(q,y)<\\epsilon\\}"

such that

"\\partial A\\cap B(q,\\epsilon)\\backslash \\{q\\}\\neq \\varphi"

but if we choose "\\epsilon" small enough ,such "\\epsilon" always exist ,we get

"\\partial A\\cap B(q,\\epsilon)\\backslash \\{q\\}= \\varphi"

Hence contradicting the hypothesis that "q" is limit point of "A" . This can be easily seen in the below rough figure


Clearly, the ball "K" inside "A" and "F" outside "A" does not intersect with boundary "\\partial A" . Hence, proved.

Thus claim1 guaranteed that if "\\partial A" has limit points then it must be on "\\partial A".



Claim2: Every point on the boundary "\\partial A" is limit point of "\\partial A"

Proof:let for any arbitrary point "p\\in \\partial A" ,thus for every "\\epsilon>0" there exist an open ball "B(p,\\epsilon)" such that

"\\partial A \\cap B(p,\\epsilon)\\backslash \\{p\\}\\neq \\varphi"

which is very clear from the above rough sketch, hence "p" is the limit point of "\\partial A".

As,"p" is arbitrary,thus every point of "\\partial A" is limit point of "\\partial A \\implies" every limit points of "\\partial A" are contained in "\\partial A" . Hence proved.


Therefore immediately from Claim 2, "\\partial A" is close set.

Hence, we are done.


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Comments

Assignment Expert
10.06.20, 22:39

Dear Brain, You are welcome. We are glad to be helpful. If you liked our service, please press a like-button beside the answer field. Thank you!

Brain
10.06.20, 12:06

Before I was totally confused about it but this clear and specially the precies writing makes me completely out of confusion now. Thank you for such a nice solution.

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