Is discrete metric compact?
A discrete space is compact if and only if it is finite. Every discrete uniform or metric space is complete. Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is finite. A finite space is metrizable only if it is discrete.
Which subsets of discrete metric space is compact?
A subset K of a metric space X is said to be compact if every open cover of K has a finite subcover. For instance, every finite set is compact; if K has the discrete metric, then K is compact if and only if it is finite (why?).
What is compact set in metric space?
Defn A set K in a metric space (X,d) is said to be compact if each open cover of K has a finite subcover. Theorem Each compact set K in a metric space is closed and bounded. Proposition Each closed subset of a compact set is also compact.
Why is discrete metric not compact?
Since K is an infinite subset of X, it follows K is an infinite discrete metric space. Consider G= {{k} | k∈K}, which is an open cover of K. Clearly, G has no finite subcover. Thus, K is not compact.
Are discrete sets countable?
Any discrete set X in R is countable. Indeed, by Proposition 1, if a set X is discrete in R, then any finite interval I contains only a finite number of points from X.
Are discrete sets closed?
Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this. “Discrete Sets and Isolated Points.” §4.6.
Are compact sets closed?
Compact sets need not be closed in a general topological space. For example, consider the set {a,b} with the topology {∅,{a},{a,b}} (this is known as the Sierpinski Two-Point Space). The set {a} is compact since it is finite.
Is the intersection of compact sets compact?
The intersection of any number of compact sets is a closed subset of any of the sets, and therefore compact.
What is compact metric?
A metric space X is compact if every open cover of X has a finite subcover. 2. A metric space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X. In fact, [0,1] is also compact (as we will see shortly).
Is the set 1 N compact?
The set N is closed, but it is not compact. The sequence (n) in N has no convergent subsequence since every subsequence diverges to infinity. As these examples illustrate, a compact set must be closed and bounded.
Are compact sets closed in metric spaces?
Theorem: Compact subsets of metric spaces are closed.
Is Z a compact?
Thus {Vi | i ∈ F} is a finite subcover of {Ui |i ∈ I} and we have shown that every open cover of Z has a finite subcover. Hence Z is compact.
How to define a discrete metric on X?
Let X ≠ ∅. Define the discrete metric on X with: d ( x, y) = { 1, x ≠ y 0, x = y (a) Ascertain the compact sets in ( X, d). I’m really confused about this tasks because I don’t know what I have to do exactly.
How to tell if a compact set is finite?
Let C be a compact set. As every set is open, for each c ∈ C the set { c } is open and ⋃ c ∈ C { c } is an open cover of C. Now, it must have a finite subcover, which tells you that C is finite.
What is the generalization of the property of compactness?
Compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line: the Heine-Borel Property. While compact may infer “small” size, this is not true in general. We will show that [0;1] is compact while (0;1) is not compact.
What is the definition of a metric space?
A metric space is a setXthat has a notion of the distanced(x,y) between every pair of pointsx,y ∈ X. The purpose of this chapter is to introduce metric spaces and give some definitions and examples. We do not develop their theory in detail, and we leave the verifications and proofs as an exercise.