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Also known as first axiom of countability
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis. That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space
X
{\displaystyle X}
is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point
x
{\displaystyle x}
in
X
{\displaystyle X}
there exists a sequence
N
1
,
N
2
,
…
{\displaystyle N_{1},N_{2},\ldots }
of neighbourhoods of
x
{\displaystyle x}
such that for any neighbourhood
N
{\displaystyle N}
of
x
{\displaystyle x}
there exists an integer
i
{\displaystyle i}
with
N
i
{\displaystyle N_{i}}
contained in
N
.
{\displaystyle N.}
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.
Contents
Examples and counterexamples
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
x
{\displaystyle x}
with radius
2
−
n
{\displaystyle 2^{-n}}
for all natural numbers
n
{\displaystyle n}
form a countable local base at
x
{\displaystyle x}
.
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
ω
1
+
1
=
[
0
,
ω
1
Properties
One of the most important properties of first-countable spaces is that given a subset
(In other words, every first-countable space is a Fréchet-Urysohn space and thus also a sequential space.) This has consequences for limits and continuity. In particular, if
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where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset
A
{\displaystyle A}
and every element
x
{\displaystyle x}
in the closure of
A
,
{\displaystyle A,}
there is a sequence in
A
{\displaystyle A}
converging to
x
.
{\displaystyle x.}
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.
f
{\displaystyle f}
is a function on a first-countable space, then
f
{\displaystyle f}
has a limit
L
{\displaystyle L}
at the point
x
{\displaystyle x}
if and only if for every sequence
x
n
→
x
,
{\displaystyle x_{n}\to x,}
where
x
n
≠
x
{\displaystyle x_{n}\neq x}
for all
n
,
{\displaystyle n,}
we have
f
(
x
n
)
→
L
.
{\displaystyle f\left(x_{n}\right)\to L.}
Also, if
f
{\displaystyle f}
is a function on a first-countable space, then
f
{\displaystyle f}
is continuous if and only if whenever
x
n
→
x
,
{\displaystyle x_{n}\to x,}
then
f
(
x
n
)
→
f
(
x
)
.
{\displaystyle f\left(x_{n}\right)\to f(x).}
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). One such space is the ordinal space
[
0
,
ω
1
)
.
{\displaystyle \left[0,\omega _{1}\right).}
Every first-countable space is compactly generated.
Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be.
