In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly...

mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X {\displaystyle X} is...

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle...

Hausdorff second-countable space is paracompact. The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor...

Lindelöf space, in particular in a second-countable space, is countable. This is proved by a similar argument as in the result above for compact spaces. A collection...

fact above about second countable scattered spaces, together with the fact that a subset of a second countable space is second countable.) Furthermore,...

theorem, second-countable then implies metrizable. Conversely, a compact metric space is second-countable. There are many natural examples of space-filling...

very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential. In any topological space ( X , τ ) , {\displaystyle...

Every regular second-countable space is completely normal, and every regular Lindelöf space is normal. Also, all fully normal spaces are normal (even...

sample space is equal to one: P ( Ω ) = 1 {\displaystyle P(\Omega )=1} . Discrete probability theory needs only at most countable sample spaces Ω {\displaystyle...

particular, every countable space is Lindelöf. A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf...

spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrizable exactly in case they are second countable....

confused with the countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the Cartesian product of countably infinitely many...

set first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable...

Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable...

Lindelöf. Every second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset). A metric space is separable...

This states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical...

set first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable...

a Gδ space is a Gδ space. Every metrizable space is a Gδ space. The same holds for pseudometrizable spaces. Every second countable regular space is a...

directed joins. Second category See Meagre. Second-countable A space is second-countable or perfectly separable if it has a countable base for its topology...

first-countable space is a Fréchet–Urysohn space. Consequently, every second-countable space, every metrizable space, and every pseudometrizable space is...

metric space is bounded. Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is...

Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense...

In mathematics, a topological space X {\displaystyle X} is said to be a Baire space if countable unions of closed sets with empty interior also have empty...

is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if...

all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a...

countable local base. Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable...

Euclidean space. For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent. Every second-countable manifold...

specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in...

mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact...