Compact Set in Metric Space

In summary, the conversation discusses the definition of D_K in relation to a compact set K in a separable metrizable space (E, \rho) and a continuous function t \mapsto f(t). It is stated that D_K is less than or equal to t if and only if the infimum of the distance between f(q) and K for q \in \mathbb{Q}\cap [0,t] is zero. This is demonstrated by approximating D_K with rational numbers and showing that the distance between f(q_r) and K goes to zero. Therefore, the infimum is zero when D_K is less than or equal to t.
  • #1
wayneckm
68
0
Hello all,Here is my question while reading a proof.

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

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

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

May someone shed some light on this? I do not understand it. Thanks very much.Wayne
 
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  • #2
It's not really clear what the domain and codomain of your function f are.
 
  • #3
Domain of [tex] f [/tex] is [tex] \mathbb{R}^{+} [/tex]
Codomain of [tex] f [/tex] is [tex] \mathbb{R} [/tex]
 
  • #4
Huh? Then I don't understand [itex]f(t)\in K[/itex]...
 
  • #5
Oops...sorry, i misunderstood the term codomain. So codomain here should be [tex] E [/tex] as stated.
 
  • #6
If [tex] D_k \leq t[/tex] then [tex] f(D_k)\in K[/tex]. We can approximate [tex] D_k[/tex] with rational numbers, and because [tex] D_k \in [0,t][/tex] we can approximate [tex] D_k[/tex] with rational numbers in [tex] \mathbb{Q}\cap [0,t][/tex] If [tex] q_r[/tex] is such a sequence converging to [tex] D_k[/tex], the distance between [tex] f(q_r)[/tex] and [tex] f(D_k)[/tex] goes to zero, which means the distance between [tex]f(q_r)[/tex] and [tex]K[/tex] must go to zero. So the infimum of the distance between [tex] f(q)[/tex] and [tex]K[/tex] for [tex] q\in \mathbb{Q}\cap [0,t][/tex] must be zero because we just found a sequence for which the distance is arbitrarily small.

This is basically the direction [tex] D_k\leq t[/tex] implies the infimum is zero.
 

What is a compact set in a metric space?

A compact set in a metric space is a subset of the metric space that is both closed and bounded. This means that every sequence of points within the set has a limit point also within the set, and the set is contained within a finite distance from any given point in the metric space.

What is the importance of compact sets in metric spaces?

Compact sets play a crucial role in analysis and topology, as they have many important properties and allow for the application of powerful theorems. They also help define the notion of convergence, continuity, and compactness in metric spaces.

How do you determine if a set is compact in a metric space?

To determine if a set is compact in a metric space, you can use the Heine-Borel theorem, which states that a subset of a metric space is compact if and only if it is closed and bounded. Alternatively, you can also check if every open cover of the set has a finite subcover.

What is the difference between a compact set and a closed set in a metric space?

A compact set is a subset of a metric space that is both closed and bounded, while a closed set is a subset of a metric space that contains all its limit points. Not all closed sets are compact, but all compact sets are closed.

What are some examples of compact sets in metric spaces?

Some examples of compact sets in metric spaces include a closed interval [a,b] on the real line, a closed disc in the plane, a sphere in 3-dimensional space, and the Cantor set. These are all subsets of their respective metric spaces that are both closed and bounded, making them compact sets.

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