- #1
jamilmalik
- 14
- 0
Homework Statement
Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##.I know the term "clopen" is not a very formal definition, at least not to my knowledge, but it does describe the two properties of the given sets: they are both open and closed.
Homework Equations
The Attempt at a Solution
Would I have to show that the metric ##\rho## satisfies the properties of a metric or would I need to show that the metric ##d## and the metric ##\rho## generate the same topology to show they are equivalent?
If I define a function ##\rho: X \times X \to \mathbb{R}_+## by the following:
if both ##x,y## lie in either ##C## or ##X \setminus C##, then ##\rho(x,y) = d(x,y)##. Otherwise, let ##\rho(x,y) = d(x,y) +100##, say. How do I proceed to show that this function satisfies the properties of a metric, that it is equivalent to ##d##, and that it satisfies the desired property of ##\rho(x,y) \geq 1 ##?
As a side thought, would using sequences be useful here, or would this be a completely different approach?
Any help with this problem is greatly appreciated as I do not know where to begin or how to proceed in writing up a correct proof.
Thanks in advance for your time and patience.