First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space ${\displaystyle X}$ is said to be first-countable if each point has a

(local base). That is, for each point ${\displaystyle x}$ in ${\displaystyle X}$ there exists a sequence ${\displaystyle N_{1},N_{2},\ldots }$ of of ${\displaystyle x}$ such that for any neighbourhood ${\displaystyle N}$ of ${\displaystyle x}$ there exists an integer ${\displaystyle i}$ with ${\displaystyle N_{i}}$ contained in ${\displaystyle N.}$ 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.

The majority of 'everyday' spaces in

centered at ${\displaystyle x}$ with radius ${\displaystyle 1/n}$ for integers form a countable local base at ${\displaystyle x.}$

An example of a space that is not first-countable is the

over an uncountable field is not first-countable.

Another counterexample is the

ordinal space

${\displaystyle \omega _{1}+1=\left[0,\omega _{1}\right]}$ where ${\displaystyle \omega _{1}}$ is the first uncountable ordinal number. The element ${\displaystyle \omega _{1}}$ is a of the subset ${\displaystyle \left[0,\omega _{1}\right)}$ even though no sequence of elements in ${\displaystyle \left[0,\omega _{1}\right)}$ has the element ${\displaystyle \omega _{1}}$ as its limit. In particular, the point ${\displaystyle \omega _{1}}$ in the space ${\displaystyle \omega _{1}+1=\left[0,\omega _{1}\right]}$ does not have a countable local base. Since ${\displaystyle \omega _{1}}$ is the only such point, however, the subspace ${\displaystyle \omega _{1}=\left[0,\omega _{1}\right)}$ is first-countable.

The quotient space ${\displaystyle \mathbb {R} /\mathbb {N} }$ 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 ${\displaystyle A}$ and every element ${\displaystyle x}$ in the closure of ${\displaystyle A,}$ there is a sequence in ${\displaystyle A}$ converging to ${\displaystyle x.}$ A space with this sequence property is sometimes called a Fréchet–Urysohn space.

First-countability is strictly weaker than

is first-countable but not second-countable.

Properties

One of the most important properties of first-countable spaces is that given a subset ${\displaystyle A,}$ a point ${\displaystyle x}$ lies in the closure of ${\displaystyle A}$ if and only if there exists a sequence ${\displaystyle \left(x_{n}\right)_{n=1}^{\infty }}$ in ${\displaystyle A}$ that converges to ${\displaystyle x.}$ (In other words, every first-countable space is a

. In particular, if ${\displaystyle f}$ is a function on a first-countable space, then ${\displaystyle f}$ has a limit ${\displaystyle L}$ at the point ${\displaystyle x}$ if and only if for every sequence ${\displaystyle x_{n}\to x,}$ where ${\displaystyle x_{n}\neq x}$ for all ${\displaystyle n,}$ we have ${\displaystyle f\left(x_{n}\right)\to L.}$ Also, if ${\displaystyle f}$ is a function on a first-countable space, then ${\displaystyle f}$ is continuous if and only if whenever ${\displaystyle x_{n}\to x,}$ then ${\displaystyle f\left(x_{n}\right)\to f(x).}$

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 ${\displaystyle \left[0,\omega _{1}\right).}$ Every first-countable space is compactly generated.

Every

of a first-countable space is first-countable, although uncountable products need not be.

• Fréchet–Urysohn space – Property of topological space
• Second-countable space – Topological space whose topology has a countable base
• Separable space – Topological space with a dense countable subset
• Sequential space – Topological space characterized by sequences

• "first axiom of countability", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• ISBN 3885380064
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