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## Main Question or Discussion Point

The question I'm doing is as follows:

(a) Show that every compact subset of a Hausdorff space is closed. I've done this.

(b) F is a compact metric space. F is a closed subset of X, and p is any point of X. Show there is a point q in F such that d(p,q)=infimum(d(p,q')) such that q' is in F.

Suppose that for every x and y in X there is a point m such that d(x,m)=0.5*d(x,y) and d(y,m)=0.5*d(x,y). Show that X is connected.

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I don't really know how to do (b). This is my attempt at the first part of (b):

Take a sequence qn in F such that d(p,qn) decreases as n increases. Then by closedness of F, the limit of qn, say q.hat, is in F. This q.hat has the property that d(p,q.hat)=inf d(p,q') as required.

But, in the above solution, I didn't use the fact that X is compact? And I can't do the second part of (b) at all.

Thanks for any help!

(a) Show that every compact subset of a Hausdorff space is closed. I've done this.

(b) F is a compact metric space. F is a closed subset of X, and p is any point of X. Show there is a point q in F such that d(p,q)=infimum(d(p,q')) such that q' is in F.

Suppose that for every x and y in X there is a point m such that d(x,m)=0.5*d(x,y) and d(y,m)=0.5*d(x,y). Show that X is connected.

--

I don't really know how to do (b). This is my attempt at the first part of (b):

Take a sequence qn in F such that d(p,qn) decreases as n increases. Then by closedness of F, the limit of qn, say q.hat, is in F. This q.hat has the property that d(p,q.hat)=inf d(p,q') as required.

But, in the above solution, I didn't use the fact that X is compact? And I can't do the second part of (b) at all.

Thanks for any help!