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

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

topological space is called countably compact if every countable open cover has a finite subcover. A topological space X is called countably compact if...

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

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...

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

-convergence in the following way. Let X {\displaystyle X} be a first-countable space and F n : X → R ¯ {\displaystyle F_{n}:X\to {\overline {\mathbb...

two simple requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities...

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

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...