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Topological space where each point has a countable neighbourhood basis
of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X {\displaystyle X}
First-countable_space
Topological space whose topology has a countable base
topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly
Second-countable_space
topological space is called countably compact if every countable open cover has a finite subcover. A topological space X is called countably compact if
Countably_compact_space
Topological space characterized by sequences
very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential. In any topological space ( X , τ ) , {\displaystyle
Sequential_space
Index of articles associated with the same name
the set first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable
Axiom_of_countability
Topological space where every sequence has a convergent subsequence
{\displaystyle n} is a sequence that has no convergent subsequence. On a first countable space, a sequence x n {\displaystyle x_{n}} has a convergent subsequence
Sequentially_compact_space
Branch of topology
the set first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable
General_topology
Type of convergence for functionals
-convergence in the following way. Let X {\displaystyle X} be a first-countable space and F n : X → R ¯ {\displaystyle F_{n}:X\to {\overline {\mathbb
Γ-convergence
Topological space with a dense countable subset
In mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle
Separable_space
Mathematical set with some added structure
analytic space Drinfeld's symmetric space Eilenberg–Mac Lane space Euclidean space Fiber space Finsler space First-countable space Fréchet space Function
Space_(mathematics)
Book by Lynn Steen
simplify the literature. For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an
Counterexamples_in_Topology
Mathematical concept
two simple requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities
Probability_space
Type of topological space
embeddings. Every first-countable space is a Fréchet–Urysohn space. Consequently, every second-countable space, every metrizable space, and every pseudometrizable
Fréchet–Urysohn_space
Topological space which is a generalization of certain compact spaces
second-countable space is paracompact. The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable
Paracompact_space
Generalization of a sequence of points
true. The spaces for which the two conditions are equivalent are called sequential spaces. All first-countable spaces, including metric spaces, are sequential
Net_(mathematics)
Mathematical function with no sudden changes
is sequentially continuous. If X {\displaystyle X} is a first-countable space and countable choice holds, then the converse also holds: any function
Continuous_function
Cluster point in a topological space
T_{1}} spaces are characterized by this property. If X {\displaystyle X} is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are)
Accumulation_point
Topological space that is homeomorphic to a metric space
states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical
Metrizable_space
Mathematical set that can be enumerated
is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if
Countable_set
Limit of some subsequence
cluster point, but not conversely. In first-countable spaces, the two concepts coincide. In a topological space, if every subsequence has a subsequential
Subsequential_limit
Property of topological spaces
Sequential spaces are CG-2. This includes first countable spaces, Alexandrov-discrete spaces, finite spaces. Every CG-3 space is a T1 space (because given
Compactly_generated_space
Relation among continuous functions
precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the
Equicontinuity
Type of vector space in math
is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is
Hilbert_space
Complement of an open subset
{\displaystyle A} also belongs to A . {\displaystyle A.} In a first-countable space (such as a metric space), it is enough to consider only convergent sequences
Closed_set
Locally convex topological vector space that is also a complete metric space
translation-invariant metric, the second a countable family of seminorms. A topological vector space X {\displaystyle X} is a Fréchet space if and only if it satisfies
Fréchet_space
Smallest ordinal number that, considered as a set, is uncountable
1 ) {\displaystyle [0,\omega _{1})} is first-countable, but neither separable nor second-countable. The space [ 0 , ω 1 ] = ω 1 + 1 {\displaystyle [0
First_uncountable_ordinal
Concept in topology
Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense
Polish_space
"Small" subset of a topological space
set (also called a meager set or a set of first category) is a subset of a topological space that is a countable union of subsets whose closures have empty
Meagre_set
Class of mathematical orderings
space is a first-countable space if and only if it has order type less than or equal to ω1 (omega-one), that is, if and only if the set is countable or
Well-order
Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable
List of general topology topics
List_of_general_topology_topics
Type of topological space
metric space is bounded. Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is
Discrete_space
Subset of Euclidean space is compact if and only if it is closed and bounded
§4.1., Exercise 7.; the reference is for a first countable space but a metric space is first countable. Bourbaki 2007, Ch. II., § 4., No. 2., Théorème
Heine–Borel_theorem
Infinite sum
is a first-countable space then it follows that the set of i ∈ I {\displaystyle i\in I} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} is countable. This
Series_(mathematics)
Property of a sequence or series
of sequences in first-countable spaces. Nets are a generalization of sequences that are useful in spaces which are not first countable. Filters further
Modes_of_convergence
Type of mathematical space
equivalent to compactness for first-countable uniform spaces). (X, d) is limit point compact (also called weakly countably compact); that is, every infinite
Compact_space
Generalization of mass, length, area and volume
{\displaystyle E\in \Sigma ,\ \ \mu (E)\geq 0} Countable additivity (or σ-additivity): For all countable collections { E k } k = 1 ∞ {\displaystyle
Measure_(mathematics)
Normed vector space that is complete
Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis
Banach_space
Generalization of "n-th" to infinite cases
are countable. Proof of first theorem: If P(α) = ∅ for some index α, then P′ is the countable union of countable sets. Therefore, P′ is countable. The
Ordinal_number
Curve whose range contains the unit square
second-countable then implies metrizable. Conversely, a compact metric space is second-countable. There are many natural examples of space-filling,
Space-filling_curve
Algebraic structure in linear algebra
are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that
Vector_space
All points and limit points in a subset of a topological space
applied to other types of closures (see below). In a first-countable space (such as a metric space), cl S {\displaystyle \operatorname {cl} S} is the
Closure_(topology)
Mathematical theorem relating to limits
is sequentially continuous. If X {\displaystyle X} is a first-countable space and countable choice holds, then the converse also holds: any function
Heine_theorem
Concept in mathematics
needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent
Axiom_of_countable_choice
Concept in topology
In mathematics, a topological space X {\displaystyle X} is said to be a Baire space if countable unions of closed sets with empty interior also have empty
Baire_space
compactness: a space (resp. a metric space) is compact if and only if each net (resp. sequence) has a cluster point. 2. For a first countable space, a point
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Mathematical space with a notion of distance
many ways: in particular, they are paracompact Hausdorff spaces (hence normal) and first-countable. The Nagata–Smirnov metrization theorem gives a characterization
Metric_space
Concept in set theory
confused with the countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the Cartesian product of countably infinitely many
Baire_space_(set_theory)
Type of topological space
Every regular second-countable space is completely normal, and every regular Lindelöf space is normal. Also, all fully normal spaces are normal (even if
Normal_space
Special subset of a partially ordered set
arbitrary directed set. In certain categories of topological spaces, such as first-countable spaces, sequences characterize most topological properties, but
Filter_(mathematics)
Vector space with generalized dot product
product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If
Inner_product_space
Type of topological space
the local properties of Euclidean space. In particular, they are locally compact, locally connected, first countable, locally contractible, and locally
Topological_manifold
Mathematical space with a notion of closeness
which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample
Topological_space
Algebraic structure of set algebra
complement, countable unions, and countable intersections. The ordered pair ( X , Σ ) {\displaystyle (X,\Sigma )} is called a measurable space. The set X
Σ-algebra
Function spaces generalizing finite-dimensional p norm spaces
-norm defined above. If I {\displaystyle I} is countably infinite, this is exactly the sequence space ℓ p {\displaystyle \ell ^{p}} defined above. For
Lp_space
Vector space of infinite sequences
{\displaystyle H} be a separable Hilbert space. Every orthogonal set in H {\displaystyle H} is at most countable (i.e. has finite dimension or ℵ 0 {\displaystyle
Sequence_space
Set of all possible outcomes or results of a statistical trial or experiment
or symbols. They can also be finite, countably infinite, or uncountably infinite. A subset of the sample space is an event, denoted by E {\displaystyle
Sample_space
Functional analysis concept
_{n}\to 0} . When the Hilbert space is in addition separable, one can mix the basis ( e n ) {\displaystyle (e_{n})} with a countable orthonormal basis for the
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Topological space that is connected
simply connected after removal of countably many points. Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such
Connected_space
In mathematics, vector space of linear forms
\mathbb {R} ^{\infty }} is countably infinite, whereas R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} does not have a countable basis. This observation
Dual_space
Class of mathematical sets
topological space X {\displaystyle X} that contains both the empty set and the entire set X {\displaystyle X} , and is closed under countable union and
Borel_set
Type of topological space in mathematics
in senses (4) or (5). The disjoint union of countably many copies of Sierpiński space is a non-compact space which is still locally compact in senses (1)
Locally_compact_space
First article on transfinite set theory
all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the
Cantor's first set theory article
Cantor's_first_set_theory_article
Space with topology generated by convex sets
separated, and countable, and the space is complete, so this metrizable space is a Fréchet space. It is known as the Schwartz space, or the space of functions
Locally convex topological vector space
Locally_convex_topological_vector_space
Type of topological space in mathematics
In mathematics, a topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle
Limit_point_compact
\operatorname {Pr} _{Y}(A).} If X {\displaystyle X} is a barreled first countable space and if C ⊆ X {\displaystyle C\subseteq X} then: If C {\displaystyle
Convex_series
Random process independent of past history
having discrete time in either countable or continuous state space (thus regardless of the state space). The system's state space and time parameter index need
Markov_chain
Generalization of closed graph, open mapping, and uniform boundedness theorem
ideally convex. Corollary—Let X {\displaystyle X} be a barreled first countable space and let C {\displaystyle C} be a subset of X . {\displaystyle X
Ursescu_theorem
Certain topology in mathematics
of the limit of the sequence, if it has one. The space ω1 is first-countable but not second-countable, and ω1+1 has neither of these two properties, despite
Order_topology
Natural number
called an involution. Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two. Two is a noun when it refers
2
Topological vector spaces
) {\displaystyle C^{\infty }(U)} can be obtained by taking a suitable countable Fréchet combination of any one of the above defining families of seminorms
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Collection of open sets used to define a topology
the real line has countable weight. If B {\displaystyle {\mathcal {B}}} is a base for the topology τ {\displaystyle \tau } of a space X {\displaystyle
Base_(topology)
Type of logical system
possible to characterize countability or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ(x)
First-order_logic
Broadest definition of sizes in integer-dimensional spaces
a way that is compatible with countable unions and other kinds of countable limits of sets. For example, every countable subset of the real line has Lebesgue
Lebesgue_measure
Line formed by the real numbers
that the topological space supports.) The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also
Number_line
Topological space with a notion of uniform properties
necessarily a metric if the space is Hausdorff. In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms
Uniform_space
Second-countable A space is second-countable or perfectly separable if it has a countable base for its topology. Every second-countable space is first-countable
Glossary_of_general_topology
sequential spaces. If the domain and/or codomain have certain additional topological properties (often, the spaces being Hausdorff and first-countable is more
Sequence_covering_map
Generalization of compactness
for every neighborhood U {\displaystyle U} of the identity and every countably infinite subset I {\displaystyle I} of S , {\displaystyle S,} there exist
Totally_bounded_space
Group that is a topological space with continuous group operations
left-invariant metric, d 0 {\displaystyle d_{0}} , as in the case of first countable spaces. By local compactness, closed balls of sufficiently small radii
Topological_group
Quotient of two integers
example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated
Rational_number
Topology made of cocountable subsets
the countable complement topology on X {\displaystyle X} , and the topological space T = ( X , T ) {\displaystyle T=(X,{\mathcal {T}})} is a countable complement
Cocountable_topology
Property of topological spaces
kleinen at any point. It is in fact totally path disconnected. A first-countable Hausdorff space ( X , τ ) {\displaystyle (X,\tau )} is locally path-connected
Locally_connected_space
of II. It is normal Haudsdorff, compact, separable, and first-countable but not second-countable. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in
Helly_space
Way of decomposing a topological space
{\displaystyle V} is a topological space (often we require that it is locally compact, Hausdorff, and second countable), S {\displaystyle {\mathcal {S}}}
Thom–Mather_stratified_space
Mathematical concept
second-countable (there are only finitely many open sets) and separable (since the space itself is countable). If a finite topological space is T1 (in
Finite_topological_space
Property of topological space
Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may
Gδ_space
In functional analysis, a Hilbert space
Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H {\displaystyle
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Generalization of boundedness
uniformly bounded. In words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Topological space
number of points. It is Hausdorff regular normal It is not: second-countable first-countable metrizable compact sequential Fréchet–Urysohn There is no sequence
Arens–Fort_space
Area of mathematical logic
characterised by properties of their type space: For a complete first-order theory T in a finite or countable signature the following conditions are equivalent:
Model_theory
Axiom of set theory
vector space with no basis. There is a vector space with two bases of different cardinalities. There is a free complete Boolean algebra on countably many
Axiom_of_choice
Topological space in mathematics
any countable ordinal α {\displaystyle \alpha } , pasting together α {\displaystyle \alpha } copies of [ 0 , 1 ) {\displaystyle [0,1)} gives a space which
Long_line_(topology)
Computational tool
In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear
Schauder_basis
On topological spaces where the intersection of countably many dense open sets is dense
sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense)
Baire_category_theorem
Set of vectors used to define coordinates
finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (non-complete) normed spaces that have countable Hamel bases. Consider
Basis_(linear_algebra)
List of concrete topologies and topological spaces
zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact. Arens square Bullet-riddled square - The space [ 0
List_of_topologies
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
index set) has a convergent subsequence if and only if there exists a countable set K ⊆ I {\displaystyle K\subseteq I} such that ( x m ) m ∈ K {\displaystyle
Bolzano–Weierstrass_theorem
Infinite cardinal number
(this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ 0
Aleph_number
Set of points on a line segment with certain topological properties
naturally homeomorphic to the countable product 2 _ N {\displaystyle {\underline {2}}^{\mathbb {N} }} of the discrete two-point space 2 _ {\displaystyle {\underline
Cantor_set
Topic in mathematics
index set I {\displaystyle I} need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This
Hilbert–Schmidt_operator
FIRST COUNTABLE-SPACE
FIRST COUNTABLE-SPACE
Girl/Female
Biblical
First-born, first fruits.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Sindhi, Tamil, Telugu
Brave; Winner; Smart; Strong; Uncountable; Infinite God
Girl/Female
British, English
First; Always First
Boy/Male
Muslim
First
Surname or Lastname
English
English : occupational name for the law-enforcement officer of a parish, from Middle English, Old French conestable, cunestable, from Late Latin comes stabuli ‘officer of the stable’. The title was also borne by various other officials during the Middle Ages, including the chief officer of the household (and army) of a medieval ruler, and this may in some cases be the source of the surname.Americanized spelling of Dutch Constapel, an occupational name for the chief gunner aboard a ship or in the garrison of a fort.
Biblical
first begotten; first fruits
Boy/Male
Shakespearean
Measure for Measure' A simple constable.
Boy/Male
Shakespearean
Love's Labours Lost' A constable.
Boy/Male
Tamil
First
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
Boy/Male
Czechoslovakian
First.
Boy/Male
Hindu, Indian
Uncountable
Boy/Male
English
From the Thicket of Trees
Boy/Male
Assamese, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
Sky; Lord of Day; Uncountable; Space
Girl/Female
Hindu
First
Girl/Female
Tamil
First
Boy/Male
Shakespearean
Much Ado About Nothing' A Constable.
Boy/Male
Norse
Pointable.
Boy/Male
Hindu, Indian
Uncountable
Girl/Female
Latin
First.
FIRST COUNTABLE-SPACE
FIRST COUNTABLE-SPACE
Boy/Male
Hindu, Indian, Parsi
Obtaining Glory
Boy/Male
Scottish
Oak.
Girl/Female
Arabic
Pillow
Boy/Male
Indian, Tamil
One who Wishes to be an Actor
Boy/Male
Arabic
Knight
Girl/Female
Hebrew
Light.
Girl/Female
Muslim/Islamic
Frail Delicate
Boy/Male
Indian, Marathi
Holy Place
Girl/Female
English
Christian.
Boy/Male
Indian, Sikh
King; Proud; Brave
FIRST COUNTABLE-SPACE
FIRST COUNTABLE-SPACE
FIRST COUNTABLE-SPACE
FIRST COUNTABLE-SPACE
FIRST COUNTABLE-SPACE
a.
Liable to be called on to render an account; answerable; as, every man is accountable to God for his conduct.
n.
A petty constable.
v. t.
To strike with the fist.
n.
The state or quality of being numerable or countable.
a.
Of the best class; of the highest rank; in the first division; of the best quality; first-rate; as, a first-class telescope.
a.
Not cogitable; inconceivable.
a.
Capable of being numbered.
v. t.
Accountable; responsible; sensitive.
a.
See Accountable.
n.
An under constable.
a.
Foremost; in front of, or in advance of, all others.
a.
Accountable.
n.
The quality or state of being accountable; accountability.
a.
Preceding all others of a series or kind; the ordinal of one; earliest; as, the first day of a month; the first year of a reign.
a.
Obtained directly from the first or original source; hence, without the intervention of an agent.
v. t.
To gripe with the fist.
a.
Most eminent or exalted; most excellent; chief; highest; as, Demosthenes was the first orator of Greece.
n.
The upper part of a duet, trio, etc., either vocal or instrumental; -- so called because it generally expresses the air, and has a preeminence in the combined effect.
adv.
Before any other person or thing in time, space, rank, etc.; -- much used in composition with adjectives and participles.