How do you show that discrete metric space is complete?

Finally, every discrete metric space is complete follows trivially from the two propositions. In a metric space, every convergent sequence is Cauchy. In a complete metric space, the converse is true, as well: every Cauchy sequence converges. “If it can converge, it does converge”.

What is complete metric space example?

If X is a topological space and M is a complete metric space, then the set Cb(X, M) consisting of all continuous bounded functions f from X to M is a closed subspace of B(X, M) and hence also complete. The Baire category theorem says that every complete metric space is a Baire space.

What is a discrete metric space?

metric space any set of points, the discrete metric specifies that the distance from a point to itself equal 0 while the distance between any two distinct points equal 1.

How do you show a metric space?

1. Show that the real line is a metric space. Solution: For any x, y ∈ X = R, the function d(x, y) = |x − y| defines a metric on X = R. It can be easily verified that the absolute value function satisfies the axioms of a metric. 2.

Why discrete metric space is complete?

Any compact metric space is sequentially compact and hence complete. The converse does not hold: for example, R is complete but not compact. In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete.

How do you show metric space is bounded?

A subset A of a metric space is called totally bounded if, for every r > 0, A can be covered by finitely many open balls of radius r. For example, a bounded subset of the real line is totally bounded. On the other hand, if ρ is the discrete metric on an infinite set X, then X is bounded but not totally bounded.

Is the discrete metric complete?

In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete.

Why is discrete metric complete?

So in discrete metric space, every Cauchy sequence is constant sequence and that way every Cauchy sequence is convergent sequence. Thus we conclude the discrete metric space is complete.

What is usual metric space?

A metric space is a set X together with such a metric. Examples. The prototype: The set of real numbers R with the metric d(x, y) = |x – y|. This is what is called the usual metric on R.

Is every discrete space Metrizable?

Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is metrizable (by the discrete metric). A finite space is metrizable only if it is discrete.

What is metric in metric space?

From Wikipedia, the free encyclopedia. In mathematics, a metric space is a non empty set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.

Is every discrete metric space complete?

How to conclude that the discrete metric space is complete?

So in discrete metric space, every Cauchy sequence is constant sequence and that way every Cauchy sequence is convergent sequence. Thus we conclude the discrete metric space is complete. Am I understanding correctly?

What is a metric space in math?

A metric space (X,d) is a set X with a metric d defined on X. is clear from the context, we refer to X as a metric space and omit explicit mention of the metric d. Example 7.2. d(x,y) = { 0 if x = y, 1 if x ̸= y.

How do you know if a space is complete?

De nition: A metric space (X;d) is complete if every Cauchy sequence in Xconverges in X (i.e., to a limit that’s in X). is Cauchy but does not converge to an element of (0;1). Example 4: The space Rn with the usual (Euclidean) metric is complete.

Is the space of continuous functions a complete metric space?

The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C (a, b) of continuous functions on (a, b), for it may contain unbounded functions.