In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".
is said to be first-countable if each point has a countable neighbourhood basis (local base).
there exists a sequence
there exists an integer
Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.
The majority of 'everyday' spaces in mathematics are first-countable.
In particular, every metric space is first-countable.
To see this, note that the set of open balls centered at
for integers form a countable local base at
An example of a space that is not first-countable is the cofinite topology on an uncountable set (such as the real line).
More generally, the Zariski topology on an algebraic variety over an uncountable field is not first-countable.
Another counterexample is the ordinal space
is the first uncountable ordinal number.
is a limit point of the subset
does not have a countable local base.
The quotient space
where the natural numbers on the real line are identified as a single point is not first countable.
[1] However, this space has the property that for any subset
A space with this sequence property is sometimes called a Fréchet–Urysohn space.
First-countability is strictly weaker than second-countability.
Every second-countable space is first-countable, but any uncountable discrete space is first-countable but not second-countable.
One of the most important properties of first-countable spaces is that given a subset
lies in the closure of
This has consequences for limits and continuity.
is a function on a first-countable space, then
is a function on a first-countable space, then
In first-countable spaces, sequential compactness and countable compactness are equivalent properties.
However, there exist examples of sequentially compact, first-countable spaces that are not compact (these are necessarily not metrizable spaces).
Every first-countable space is compactly generated.
Any countable product of a first-countable space is first-countable, although uncountable products need not be.