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...

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

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...

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

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...

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

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...

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

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

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

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

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...

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...

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

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

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...

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

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

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...

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

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...

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...

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...

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

translation-invariant metric, the second a countable family of seminorms. A topological vector space X {\displaystyle X} is a Fréchet space if and only if it satisfies...

are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that...

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