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yes

sorry I was working on it, didn't see your comment..

we know A is closed because of definition, right..

A is closed when A=cl(A).

in (a) u r supposed to prove that cl(A) is closed.

I meant cl(A) is closed, sory

that is what u r supposed to prove.

listen. i have to go.
i'll try to log-in later.
good luck with this.

thanks, but due is tomorrow, hurry up (:

quick question

I know how to prove in R, is it the same in M

or if the theorem valid for R, can we use for M ( metric space)

@TuringTest any idea?

all new to me, sorry
sounds cool though!

it is (:

(a) we have to prove \(cl(cl(A))=cl(A)\).
let \(x_0\in cl(cl(A))\), this means that for every \(\varepsilon>0\)
\[ \large B(x_0,\varepsilon)\cap cl(A)\neq\emptyset. \]
Question: \(B(x_0,\varepsilon)\cap A\neq\emptyset\) ?
Let \(x_1\in B(x_0,\varepsilon)\cap cl(A)\) and \(x_1\neq x_0\). Let \(\delta

wow man nice job, thanks a lot..

u r welcome.
it was fun remembering all this.

What math is this please?

It is mixed of topology and real analysis, in particularly, closed set, neighborhood