Stuck in compactness, connectedness.

Question

I want to confirm my answer

The question is:- Find a closed set in R which is neither compact nor connected.

Can I write {1} union [0,infinity) ?

Next I am in search of example of a function f: X--> Y such that X is connected metric space but f(X) is disconnected subset of metric space Y. I know R is connected metric space with usual metric

Do you mean ${-1}$ instead of ${1}$? – Oct 28 '15 at 07:12 — , Oct 28 '15 at 07:12
Yes it is {-1}. Sorry for the mistake. – Kavita Oct 28 '15 at 07:33 — Kavita, Oct 28 '15 at 07:33

score 2 · Accepted Answer · answered Oct 28 '15 at 07:21

2

You probably mean $\{-1\} \cup [0, \infty)$, as otherwise you are just left with $[0,\infty)$, as $1 \in [0, \infty)$. Then it is OK.

For your $f$, use your suggested $X$ and let $Y$ be the reals as well. Send $0$ to some point, all other points to some other point.

Note that $f$ must be discontinuous (as $f$ continuous implies that the image will be connected).

answered Oct 28 '15 at 07:21

Henno Brandsma

242,131

Can I have f: R-->R defined by f(x)= 1 if x=0 . and f(x)=2 if x is in R*? – Kavita Oct 28 '15 at 08:06
Sure you can. Why not? – Henno Brandsma Oct 28 '15 at 08:07

Stuck in compactness, connectedness.

1 Answers1