Doubt regarding open set

ashok vardhan

Sir , we say that in an open set U for every point (x0,y0) there exists some r>0 such that B((x0,y0),r) lies in U.. where B stands for open disk around (x0,y0) with radiuus r...

My doubt is does there exist some "p" such that closed disk around (x0,y0) with radius "p" lies in the open set U...is this possible for all open sets..or not at all...please clarify me....

Thanking You.

micromass

Yes, what you say is indeed true. And it's not hard to prove:

Given an open set U and a point x in U. We can find an open ball [itex]B(x,r)[/itex] in U. But then the closed ball [itex]B^*(x,r/2)[/itex] is easily seen to be a subset of [itex]B(x,r)[/itex] and thus of U.

So indeed, for every open U and every x in U, we can find a closed ball [itex]B^*(x,p)\subseteq U[/itex].

I have to warn you: this is true in metric spaces. But it is not necessarily true in topological spaces (if you replace "ball" with a suitable other notion).