How Does Compactness Affect Function Behavior in Metric Spaces?

[wayneckm]

Hello all,Here is my question while reading a proof.

For a compact set K in a separable metrizable spce (E,\rho) and a continuous function t \mapsto f(t), if we define

D_{K} = \inf \{ t \geq 0 \; : \; f(t) \in K \}

then, D_{K} \leq t if and only if \inf\{ \rho(f(q),K) : q \in \mathbb{Q} \cap [0,t] \} = 0

May someone shed some light on this? I do not understand it. Thanks very much.Wayne

 

[Landau]

It's not really clear what the domain and codomain of your function f are.

 

[wayneckm]

Domain of f is \mathbb{R}^{+}
Codomain of f is \mathbb{R}

 

[Landau]

Huh? Then I don't understand f(t)\in K...

 

[wayneckm]

Oops...sorry, i misunderstood the term codomain. So codomain here should be E as stated.

 

[Office_Shredder]

If D_k \leq t then f(D_k)\in K. We can approximate D_k with rational numbers, and because D_k \in [0,t] we can approximate D_k with rational numbers in \mathbb{Q}\cap [0,t] If q_r is such a sequence converging to D_k, the distance between f(q_r) and f(D_k) goes to zero, which means the distance between f(q_r) and K must go to zero. So the infimum of the distance between f(q) and K for q\in \mathbb{Q}\cap [0,t] must be zero because we just found a sequence for which the distance is arbitrarily small.

This is basically the direction D_k\leq t implies the infimum is zero.