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

## 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.