Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, that is, separated. A discrete space is compact if and only if it is finite. Every discrete uniform or metric space is complete.

Is a discrete space locally compact?

Any discrete space X is locally compact. Note that X is compact if and only if X is finite. Example 1.4. Euclidean space Rn is locally compact.

Which discrete spaces are compact?

For example, if X is a discrete space (every subset is open) then X is compact if and only if X is a finite set (if you had an infinite discrete space then the collection {{x} : x∈X} is an open cover with no finite subcover).

Is discrete topology Metrizable?

Indiscrete Topology is not Metrizable.

What is a discrete set topology?

discrete set (plural discrete sets) (topology) A set of points of a topological space such that each point in the set is an isolated point, i.e. a point that has a neighborhood that contains no other points of the set.

Lecture 66 | if X is infinite and discrete then X is not compact

19 related questions found

Is an infinite discrete space compact?

Any infinite discrete space is not compact. In fact, if X is an infinite discrete space, then {{x} | x ∈ X} is an open cover of X which has no finite subcover.

Are all 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.

Is a discrete set 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.

How do you prove discrete topology?

Discrete Topology is Topology

1. Let S be a set.
2. Let τ be the discrete topology on S.
3. Let T=(S,τ) be the discrete space on S.
4. Then by definition τ=P(S), that is, is the power set of S.
5. We confirm the criteria for T to be a topology:
6. ◼

Is discrete topological space connected?

Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space. ... The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected.

Are all closed sets compact?

every compact set is closed, but not conversely. There are, however, spaces in which the compact sets coincide with the closed sets-compact Hausdorff spaces, for example.

Are closed intervals compact?

The bounded closed interval [0, 1] is compact and its maximum 1 and minimum 0 belong to the set, while the open interval (0, 1) is not compact and its supremum 1 and infimum 0 do not belong to the set. The unbounded, closed interval [0, ∞) is not compact, and it has no maximum.

Are all closed and bounded sets compact?

By the Heine-Borel Theorem, closed and bounded subsets in Rn are compact, but this fact does not hold in metric spaces—even complete metric spaces— generally. We give necessary and sufficient conditions on a metric space for closed and bounded subsets to be compact.

Is a closed subset of a compact set compact?

37, 2.35] A closed subset of a compact set is compact. Proof : Let K be a compact metric space and F a closed subset. Then its complement Fc is open. ... Since K is compact, Ω has a finite subcover; removing Fc if necessary, we obtain a finite subcollection of {Vα} which covers F.

Why is Q not locally compact?

Since Q is dense in R and Q⊆F, it follows that F=R. But then U=Q, and we know that Q is not open in R! Therefore Q is not locally-compact.

Is L2 locally compact?

Some common spaces are not locally compact, such as infinite dimensional Banach spaces. For instance, the L2-space of square integrable functions is not locally compact.

Is every discrete set at most countable?

Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of S may be mapped into a set of points with rational coordinates, of which there are only countably many.

Which of the following is discrete topology?

set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X.

Is discrete metric space open or closed?

As any union of open sets is open, any subset in X is open. Now for every subset A of X, Ac = X\A is a subset of X and thus Ac is a open set in X. This implies that A is a closed set. Thus every subset in a discrete metric space is closed as well as open.

Can a compact set have an isolated point?

Yes. Take X={0}∪{2−n:n∈N} with the topology given by the distance d(x,y)=|x−y|. Then X is compact (it is bounded and closed) and it contains infinitely many isolated points, namely {2−n:n∈N}.

Is an isolated point open or closed?

An isolated point is closed (no limit points to contain). A finite union of closed sets is closed. Hence every finite set is closed.

Is an isolated point a interior point?

A point s S is called interior point of S if there exists a neighborhood of s completely contained in S. The set of all interior points of S is called the interior, denoted by int(S). A point t S is called isolated point of S if there exists a neighborhood U of t such that U S = { t }.

What sets are discrete?

A discrete space is one in which the single point sets are defined to be open. If, for S={1,2,3} we declare: ∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3} to all be open sets then that is a discrete space.

Are discrete spaces 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.

What makes something discrete?

Discrete data is information that can only take certain values. ... Height, weight, temperature and length are all examples of continuous data.