Mathematics Metric Spaces Problems

Description

1. Let M is a metric space. Let A ? M be dense in M, i.e., ClosMA = M. Suppose that f : M ? R is continuous and the restriction f|A : A ? R is uniformly continuous.
(i) Showthatforeveryx?M andevery?>0,thereexistsx? ?Asuchthat |f(x) ? f(x?)| < ? and d(x, x?) < ?.
(ii) Show that f is uniformly continuous.
2. Consider the metric space (M,d) where M = {x ? R|0 ? x ? 1} with d(x,y) =
|x ? y| for x, y ? M . Let A ? M . Show that the following statements are equivalent.
(a) A is open in R.
(b) AisopeninM and0,1?/A.

(7 points) Let M is a metric space. Let A CM be dense in M, i.e., ClosmA = M. Suppose
that f:M +R is continuous and the restriction f|A: A + R is uniformly continuous.
(i) Show that for every x E M and every a 0, there exists x’ E A such that
f(x) – f(x)
n->
n->
So x E ClosMA.
Let M be a metric space. Let A CM. Define
IntMA= {x E Al 3r > 0 s.t. BM(x,r) C A},
and
ClosMA= {x E M| X
{x E M|x = lim In for some
convergent (xn)neN in A}.
n-

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