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In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.More explicitly, a topological space is second-countable if there exists some countable collection = {} = of open subsets of such that any open subset of can be written as a union of elements of some subfamily of .
sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set; first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base; separable space: there exists a countable dense subset
second-countable space: the topology has a countable base; separable space: there exists a countable dense subspace; Lindelöf space: every open cover has a countable subcover; σ-compact space: there exists a countable cover by compact spaces; Relations: Every first countable space is sequential. Every second-countable space is first-countable ...
Any second-countable space is separable: if {} is a countable base, choosing any from the non-empty gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf.
As an example we will define the T 2 axiom, which is the condition imposed on separated spaces. Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets {x} and {y} are separated by neighbourhoods. Separated spaces are usually called Hausdorff spaces or T 2 spaces.
Every metrizable space is a G δ space. The same holds for pseudometrizable spaces. Every second countable regular space is a G δ space. This follows from the Urysohn's metrization theorem in the Hausdorff case, but can easily be shown directly. [3] Every countable regular space is a G δ space. Every hereditarily Lindelöf regular space is a ...