Is compact space separable?

Proposition 2.3 Every totally bounded metric space (and in particular every compact met- ric space) is separable. Intuitively, a separable space is one that is “well approximated by a countable subset”, while a compact space is one that is “well approximated by a finite subset”.

Table of Contents

Which spaces are separable?

Separable spaces

Every compact metric space (or metrizable space) is separable.
Any topological space that is the union of a countable number of separable subspaces is separable.
The space of all continuous functions from a compact subset to the real line.

Is every locally compact metric space separable?

Every compact metric space is separable. Therefore a locally compact metric space is separable iff it is compact.

Which space is Metrizable?

It states that a topological space is metrizable if and only if it is regular, Hausdorff and has a σ-locally finite base. A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets. For a closely related theorem see the Bing metrization theorem.

Is the space of continuous functions separable?

general topology – The space of bounded continuous functions are not separable – Mathematics Stack Exchange.

Is l2 space separable?

The space l2 is much larger than any of the finite-dimensional Hilbert spaces Fn — for instance, it is not locally compact — but it is still small enough to be “separable”; this in fact topologically characterizes l2.

How do you show a space is separable?

We say a metric space is separable if it has a countable dense subset. Using the fact that any point in the closure of a set is the limit of a sequence in that set (yes?) it is easy to show that Q is dense in R, and so R is separable. A discrete metric space is separable if and only if it is countable.

Are all metric spaces compact?

Metric spaces (X, d) is compact. (X, d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces). (X, d) is sequentially compact; that is, every sequence in X has a convergent subsequence whose limit is in X (this is also equivalent to compactness for first-countable uniform spaces).

Is every closed and bounded set is compact?

In and a set is compact if and only if it is closed and bounded. In general the answer is no. There exists metric spaces which have sets that are closed and bounded but aren’t compact. Theorem 2: There exists a metric space that has a closed and bounded set that is not compact.

What is compact Hausdorff space?

A compact Hausdorff space or compactum, for short, is a topological space which is both a Hausdorff space as well as a compact space. This is precisely the kind of topological space in which every limit of a sequence or more generally of a net that should exist does exist (this prop.) and does so uniquely (this prop).

Is every normal space metrizable?

Every second countable regular space is metrizable. While every metrizable space is normal (and regular) such spaces do not need to be second countable. For example, any discrete space X is metrizable, but if X consists of uncountably many points it does not have a countable basis (Exercise 4.10).

Is discrete space separable?

A discrete space is separable if and only if it is countable. Any topological subspace of. (with its usual Euclidean topology) that is discrete is necessarily countable.

When the measure space is separable?

The measure ^ is called separable whenever the metric space (2(^t), d) is separable. It is a classical result that ju is separable if and only if the Banach space L'(ju) is separable [8, p. 137]. To exhibit non-separable measures is not a problem; see [8, p.

Are all metric spaces separable?

Abstract. We first show that in the function realizability topos RT(K2) every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every T0-space is separable and every discrete space is countable.

Why are separable spaces called separable?

Maybe, it refers to the separation of points by means of countable collection of small open sets (where separation means that for any two points we may find their disjoint neighborhoods in our collection). It is probably not formally equivalent to the existence of a countable dense subset, but something similar.

What is a compact metric space?

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. 10.3 Examples.

Is compact set bounded?

Theorem A compact set K is bounded. Proof Pick any point p ∈ K and let Bn(p) = {x ∈ K : d(x, p) < n}, n = 1,2,…. These open balls cover K. By compactness, a finite number also cover K.

Is a compact metric space bounded?

We start with the fact that in any metric space, a compact subset is closed and bounded. Bounded here means that the subset “does not extend to infinity,” that is, that it is contained in some open ball around some point.

Are compact Hausdorff spaces metrizable?

A Hausdorff compact space is metrizable if and only if it is a continuous open image of the Sorgenfrey line. In this note we prove that a regular continuous open image of the Sorgenfrey line with an uncountable weight has a closed subspace that is homeomorphic to the Sorgenfrey line.

Is every compact space Hausdorff?

Compact subsets of Hausdorff spaces are closed and closed subsets of compact spaces are compact. There are compact but non Hausdorff spaces, and likewise there are Hausdorff but not compact spaces.

Is every compact metric space separable?

Every compact metric space (or metrizable space) is separable. Any topological space that is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that

What is a separable space in math?

Separable spaces Every compact metric space (or metrizable space) is separable. Any topological space which is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that n {\\displaystyle n} -dimensional Euclidean space is separable.

Is-dimensional Euclidean space separable?

What is the difference between metrizable and second countable spaces?

Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf . To further compare these two properties: