## Is every open subspace of a locally compact space is locally compact?

Every open topological subspace X⊂openK of a compact Hausdorff space K is a locally compact topological space. In particular every compact Hausdorff space itself is locally compact.

## Is compact subset open?

So unlike with closed and open sets, a set is “compact relative a subset Y ” if and only if it is compact relative to the whole space. Compact subsets of a metric space are closed. Closed subsets of compact sets are compact. If F is closed and K is compact then F ∩ K is compact.

**Which space are locally compact?**

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.

### Is locally compact Hausdorff space normal?

A locally compact Hausdorff space is always locally normal. A normal space is always locally normal. A T1 space need not be locally normal as the set of all real numbers endowed with the cofinite topology shows.

### 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. Thus if {Vα} is an open cover of F we obtain an open cover Ω of K by adjoining Fc.

**Is every metric space locally compact?**

The answer is no: for instance, if X is any discrete metric space, then every real-valued function on X is automatically both continuous and uniformly continuous (why?); but a discrete metric space is compact if and only if it is finite (why?).

## Are all compact sets closed?

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. It is the intent of this note to give several characterizations of such spaces and to list some of their properties.

## Is a locally compact space compact?

Definition. A topological space is locally compact if every point has a neighborhood base consisting of compact subspaces.

**Is every compact set 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 every subset of compact space compact?

Every closed subspace of a compact space is compact. Proof. Let Y be a closed subspace of the compact space X. Given a covering A of Y by sets open in X, let us form an open covering B of X by adjoining to A the single open set X − Y , that is, B = A∪{X − Y }.

### What is open set in metric space?

In a metric space—that is, when a distance function is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

**Is Cantor set compact?**

The Cantor ternary set, and all general Cantor sets, have uncountably many elements, contain no intervals, and are compact, perfect, and nowhere dense.