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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 .
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. To further compare these two properties: An arbitrary subspace of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
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
Every Polish space is second countable (by virtue of being separable and metrizable). [1]A subspace Q of a Polish space P is Polish (under the induced topology) if and only if Q is the intersection of a sequence of open subsets of P (i. e., Q is a G δ-set).
{p} is dense and hence X is a separable space. However if X is uncountable then X \ {p} is not separable. This is an example of a subspace of a separable space not being separable. Countability (first but not second) If X is uncountable then X is first countable but not second countable. Alexandrov-discrete The topology is an Alexandrov topology.
For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent. Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components. In particular, a connected ...
A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf, [5] but not conversely. For example, there are many compact spaces that are not second-countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. [6] Every regular Lindelöf space ...
The sets and are separated by a continuous function if there exists a continuous function: from the space to the real line such that () and (), that is, members of map to 0 and members of map to 1. (Sometimes the unit interval [ 0 , 1 ] {\displaystyle [0,1]} is used in place of R {\displaystyle \mathbb {R} } in this definition, but this makes ...