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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 h ...
, a second-countable space, also called a completely separable 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 po ...
whose topology 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 ...
base. More explicitly, a topological space T is second-countable if there exists some countable collection \mathcal = \_^ of
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
subsets of T such that any open subset of T can be written as a union of elements of some subfamily of \mathcal. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many "
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. T ...
" spaces in mathematics are second-countable. For example,
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
(R''n'') with its usual topology is second-countable. Although the usual base 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 defin ...
s is
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
, one can restrict to the collection of all open balls with
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.


Properties

Second-countability is a stronger notion than first-countability. A space is first-countable if each point has a countable
local base 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 Neighbou ...
. Given a base for a topology and a point ''x'', the set of all basis sets containing ''x'' forms a local base at ''x''. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
is first-countable but not second-countable. Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
subset) and Lindelöf (every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
has a countable subcover). The reverse implications do not hold. For example, the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...
on the real line is first-countable, separable, and Lindelöf, but not second-countable. For
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 set ...
s, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.Willard, theorem 16.11, p. 112 Therefore, the lower limit topology on the real line is not metrizable. In second-countable spaces—as in metric spaces— compactness, sequential compactness, and countable compactness are all equivalent properties.
Urysohn's metrization theorem In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
states that every second-countable, Hausdorff
regular space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can b ...
is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
. It follows that every such space is completely normal as well as
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norma ...
. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.


Other properties

*A continuous,
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of a second-countable space is second-countable. *Every subspace of a second-countable space is second-countable. * Quotients of second-countable spaces need not be second-countable; however, ''open'' quotients always are. *Any countable
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of a second-countable space is second-countable, although uncountable products need not be. *The topology of a second-countable T1 space has
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
less than or equal to ''c'' (the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
). *Any base for a second-countable space has a countable subfamily which is still a base. *Every collection of disjoint open sets in a second-countable space is countable.


Examples and counterexamples

* Consider the disjoint countable union X = ,1\cup ,3\cup ,5\cup \dots \cup k, 2k+1\cup \dotsb. Define an equivalence relation and a
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on. ''X'' is second-countable, as a countable union of second-countable spaces. However, ''X''/~ is not first-countable at the coset of the identified points and hence also not second-countable. * The above space is not homeomorphic to the same set of equivalence classes endowed with the obvious metric: i.e. regular Euclidean distance for two points in the same interval, and the sum of the distances to the left hand point for points not in the same interval -- yielding a strictly coarser topology than the above space. It is a separable metric space (consider the set of rational points), and hence is second-countable. * The
long line Long line or longline may refer to: *'' Long Line'', an album by Peter Wolf *Long line (topology), or Alexandroff line, a topological space * Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...
is not second-countable, but is first-countable.


Notes


References

* Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts. * John G. Hocking and Gail S. Young (1961). ''Topology.'' Corrected reprint, Dover, 1988. {{Topology General topology Properties of topological spaces