first-countable space

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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a first-countable space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
satisfying the "first axiom of countability". Specifically, a space $X$ is said to be first-countable if each point has a
countable In mathematics, a set 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 there exists an injective function from it into the natural numbers; ...
neighbourhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
(local base). That is, for each point $x$ in $X$ there exists a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
$N_1, N_2, \ldots$ of neighbourhoods of $x$ such that for any neighbourhood $N$ of $x$ there exists an integer $i$ with $N_i$ contained in $N.$ Since every neighborhood of any point contains an open neighborhood of that point, the
neighbourhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
can be chosen
without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
to consist of open neighborhoods.

# Examples and counterexamples

The majority of 'everyday' spaces in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
are first-countable. In particular, every
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
is first-countable. To see this, note that the set of
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defi ...
s centered at $x$ with radius $1/n$ for integers form a countable local base at $x.$ An example of a space which is not first-countable is the
cofinite topology In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocou ...
on an uncountable set (such as the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
). Another counterexample is the ordinal space
first uncountable ordinal In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. ...
number. The element $\omega_1$ is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
of the subset even though no sequence of elements in $\left\left[0, \omega_1\right\right)$ has the element $\omega_1$ as its limit. In particular, the point $\omega_1$ in the space

# Properties

One of the most important properties of first-countable spaces is that given a subset $A,$ a point $x$ lies in the closure of $A$ if and only if there exists a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
$\left\left(x_n\right\right)_^$ in $A$ which converges to $x.$ (In other words, every first-countable space is a Fréchet-Urysohn space and thus also a sequential space.) This has consequences for
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
and continuity. In particular, if $f$ is a function on a first-countable space, then $f$ has a limit $L$ at the point $x$ if and only if for every sequence $x_n \to x,$ where $x_n \neq x$ for all $n,$ we have $f\left\left(x_n\right\right) \to L.$ Also, if $f$ is a function on a first-countable space, then $f$ is continuous if and only if whenever $x_n \to x,$ then $f\left\left(x_n\right\right) \to f\left(x\right).$ In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily not metrizable spaces). One such space is the ordinal space
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of a first-countable space is first-countable, although uncountable products need not be.