Connectedness of Subsets in Metric Spaces: A Math Problem

[gotjrgkr]

Homework Statement

Let X and Y are metric spaces where X contains Y, and A is a subset of Y
;that is,
{Let X be a metric space with the metric d(x,y).
Let Y be a nonempty subset of X. Then we can regard Y as a metric space with
the same metric d(x,y).
Let A be a subset of Y. (Then, A is also a subset of X). }.

is it true that A is connected relative to Y if and only if A is connected
relative to X? I know it is true for compactness.
;
My question is, are the following two conditions equivalent? (in other
words, is it an if and only if statement?)
a) A is a connected subset of Y when Y is regarded as a metric space with
the metric d(x,y).
b) A is a connected subset of X with the metric d(x,y).

 

[HallsofIvy]

Yes, those are equivalent. Since "connectedness" is defined "negatively' (a set is connected if and only if it is NOT the union of connected sets), it is simplest to use proof by contradiction. Suppose A is connected relative to Y but not connected relative to X. Then there exist two separated (in X) sets, U and V, such that $A= U\cup V$. Separated "in X" means that $\overline{U}\cap V$ and $U\cap\overline{V}$ are empty where the closure is in X. Take the intersection of U and V with Y to get the corresponding sets in Y and look at their closures in Y.

Suppose A is connected in X but not in Y. Basically, do the same thing in reverse.

 

[gotjrgkr]

Thank you so much...!
It helps me a lot.