Question

Show directly that a compact metric space is totally bounded

Answer #1

These problems concern the discrete metric. You can assume that
the underlying space is R.
(a) What does a convergent sequence look like in the discrete
metric?
(b) Show that the discrete metric yields a counterexample to the
claim that every bounded sequence has a convergent subsequence.
(c) What does an open set look like in the discrete metric? A
closed set?
(d) What does a (sequentially or topologically) compact set look
like in the discrete metric?
(e) Show that...

(4)
Show that a totally bounded set is bounded. Is the converse
true?

Let (X, d) be a compact metric space and let A ⊆ X. Suppose that
A is not compact. Prove that there exists a continuous function f :
A → R, from (A, d) to (R, d|·|), which is not uniformly
continuous.

Prove that a continuous real-valued function on a compact metric
space assumes its maximum value.
I want to show this. Pleas explain step by step

4. Is complete metric space always compact? Answer the question
by considering the real line R. Justify your
argument in detail.

Let (X,d) be a metric space which contains an infinite countable
set Ewith the property x,y ∈ E ⇒ d(x,y) = 1.
(a) Show E is a closed and bounded subset of X. (b) Show E is
not compact.
(c) Explain why E cannot be a subset of Rn for any n.

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

Supply proofs for the following miscellaneous propositions from
the course in a metric space context:
(e) A set is open if and only if its complement is closed.
(f) A compact set (you may use either definition) is closed and
bounded.

Let A⊆(X,d) a metric space. Suppose there are an infinite number
of elements in e1,e2,e3,...∈ A such that d(ei,ej)=4 if i≠j and
d(ei,ej)=0 if i=j for i,j=1,2,3...
Prove that A is not totally bounded.
(Please do not write in script and show all your steps and
definitions used)

A topological space is totally disconnected if the connected
components are all singletons. Prove that any countable metric
space is totally disconnected.

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