Finding an unbounded set with a specific property

Question

Find an unbounded subset $A ⊂ \Bbb R$ such that every function from $A$ to a metric space is uniformly continuous.

My attempt at the solution (incomplete).

If $A⊆ \Bbb R$ were such a set, then for each $x_{0} ∈A$ the function $f_{x_{0}}:A→\Bbb R$ defined by $f_{x_{0}}(x)=\{1$,if $x=x_{0}$

        $0,if $ x≠x_{0} $

is continuous.

I don't know how to proceed further. Please suggest.

John · Answer 1 · 2014-11-12T06:44:19.493

2

Hint:

Suppose A has discrete metric, then every function frome it to a metric space is uniformly continuous.
Now you are asked to find a set $A\subset \mathbb{R}$, such that every function from to metric space $Y$ it is uniformly continuous in terms of the subspace topology (This is what $f:A\to Y$ is uniformly continuous means). Note the induced subspace topology on a subset of $\mathbb{R}$ can have discrete metric even if $\mathbb{R}$ has standard metric. Now can you find an unbounded subset of $\mathbb{R}$ whose subspace topology is discrete metric?

$\mathbb{N}\subset \mathbb{R}$ has discrete metric.

edited Nov 12 '14 at 06:44

answered Nov 12 '14 at 06:23

John

No I am not thinking to use it. My question is more general for the usual metric on $\Bbb R$ – Silver moon Nov 12 '14 at 06:26
@Martin So what's your topology on R? – John Nov 12 '14 at 06:28
Usual distance metric $|x-y|$ – Silver moon Nov 12 '14 at 06:29
@Martin OK. Then which part of my answer do you have question? – John Nov 12 '14 at 06:33
I just don't want to use discrete metric space, can it be done without this assumption? – Silver moon Nov 12 '14 at 06:36
Can you please elaborate a bit on your 2nd part as well? I do not understand it. – Silver moon Nov 12 '14 at 06:38
@Martin I have added more details. – John Nov 12 '14 at 06:45
Please give a definition of "the induced subspace topology". Also since I have not read about this (I am doing first course in analysis), can you please explain "the induced subspace topology on a subset of R can have discrete metric even if R has standard metric"-How? – Silver moon Nov 12 '14 at 07:01
@Martin Here is a definition for metric subspace. And to show a subspace of $\mathbb{R}$ can have discrete metric, see here. – John Nov 12 '14 at 07:11
Hi, I have not been taught about metric subspace yet. So, I think I cannot use that now, http://math.stackexchange.com/questions/356962/metric-space-and-uniformly-continuous-functions I found this link but I do not seem to understand this. Can you help me with this proof? – Silver moon Nov 12 '14 at 07:18

1 Answers1