Is every first-countable space metrizable?

Is every first-countable space metrizable?

Every metric space is first-countable. For x ∈ X, consider the neighborhood basis Bx = {Br(x) | r > 0,r ∈ Q} consisting of open balls around x of rational radius.

Is every Hausdorff space first countable?

It is well known that every first countable, countably compact, Hausdorff space is regular. Theorem 2.13. Let 2P denote any one of paracompact, weakly normal, normal, and completely normal, and let (X, S~) be a first countable S-space.

What is first countable in topology?

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the “first axiom of countability”. Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base).

Does first countable implies second countable?

Second-countability is a stronger notion than first-countability. A space is first-countable if each point has a countable local base. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space.

Is compact space first countable?

A space X is first countable if every point has a countable local base; separable if X has a countable dense set; locally compact if every point has a compact neighborhood; and zero-dimensional if every point has a local base of clopen sets.

Is every Metrizable space normal?

Exactly the same proof shows that every metrizable space is normal.

Are the real numbers first countable?

In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities.

Is cofinite topology compact?

Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology. Compactness: Since every open set contains all but finitely many points of X, the space X is compact and sequentially compact. If X is finite then the cofinite topology is simply the discrete topology.

Is cofinite topology first countable?

The cofinite topology on R is finer, but is not first countable. (xix) A subspace of a second countable space is second countable. True. Hint: If Y ⊆ X and B is a countable basis for X, consider {B ∩ Y | B ∈ B}.

Is every second countable 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 the real line second countable?

The real line is the second countable space. Any uncountable set X with a countable topology is not the first countable and so is not the second countable space.

How do you prove a 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.