In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 points ...
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
is second-countable if there exists some countable collection
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
such that any open subset of
can be written as a union of elements of some subfamily of
. A second-countable space is said to satisfy the second axiom of countability. Like other
countability axiom In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist.
Important examples
Important c ...
s, 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. Th ...
" 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 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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
(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 numb ...
, 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
Neighbour ...
. 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. Mathematically ...
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\s ...
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 in ...
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 settin ...
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
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, 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 ...
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
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...
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 normal, ...
. 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 T
1 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
. 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 fish ...
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