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Topological space whose topology has a countable base
In 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 where each point has a countable neighbourhood basis
mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X {\displaystyle X} is
First-countable_space
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
Type of topological space
particular, every countable space is Lindelöf. A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf
Lindelöf_space
Index of articles associated with the same name
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 concept
Lindelöf space, in particular in a second-countable space, is countable. This is proved by a similar argument as in the result above for compact spaces. A collection
Locally_finite_collection
Topological space which is a generalization of certain compact spaces
Hausdorff second-countable space is paracompact. The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor
Paracompact_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
Concept in mathematics
Lindelöf. Every second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset). A metric space is separable
Axiom_of_countable_choice
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
Curve whose range contains the unit square
theorem, second-countable then implies metrizable. Conversely, a compact metric space is second-countable. There are many natural examples of space-filling
Space-filling_curve
fact above about second countable scattered spaces, together with the fact that a subset of a second countable space is second countable.) Furthermore,
Scattered_space
Mathematical concept
measure of the entire sample space is equal to one: P ( Ω ) = 1 {\displaystyle P(\Omega )=1} . For a countable sample space Ω {\displaystyle \Omega } ,
Probability_space
Collection of open sets used to define a topology
spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrizable exactly in case they are second countable.
Base_(topology)
Concept in set theory
confused with the countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the product of countably infinitely many copies
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
Normal_space
Topological space that is homeomorphic to a metric space
This states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical
Metrizable_space
Property of topological space
a Gδ space is a Gδ space. Every metrizable space is a Gδ space. The same holds for pseudometrizable spaces. Every second countable regular space is a
Gδ_space
Branch of topology
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
countable union of open intervals. Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover
Lindelöf's_lemma
space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is
Countably quasi-barrelled space
Countably_quasi-barrelled_space
Concept in topology
Polish space is a separable, completely metrizable topological space; i.e., a space homeomorphic to a complete metric space that has a countable dense
Polish_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
Type of topological space
Euclidean space. For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent. Every second-countable manifold
Topological_manifold
vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again
Countably_barrelled_space
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
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
Generalization of mass, length, area and volume
∅ ) = 0 {\displaystyle \mu (\varnothing )=0} Countable additivity (or σ-additivity): For all countable collections { E k } k = 1 ∞ {\displaystyle
Measure_(mathematics)
Type of mathematical space
is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete
Compact_space
"Small" subset of a topological space
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 interior. Thus meager
Meagre_set
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
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
directed joins. Second category See Meagre. Second-countable A space is second-countable or perfectly separable if it has a countable base for its topology
Glossary_of_general_topology
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
Mathematical property of a space
countable local base. Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable
Topological_property
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
List of concrete topologies and topological spaces
countable. Cofinite topology Double-pointed cofinite topology Ordinal number topology Pseudo-arc Ran space Tychonoff plank Discrete two-point space −
List_of_topologies
Type of topological space
first-countable space is a Fréchet–Urysohn space. Consequently, every second-countable space, every metrizable space, and every pseudometrizable space is
Fréchet–Urysohn_space
Type of topological space
mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact
Σ-compact_space
Vector space on which a distance is defined
space C ∞ ( K ) , {\displaystyle C^{\infty }(K),} as defined in the article on spaces of test functions and distributions, is defined by a countable family
Normed_vector_space
Book by Lynn Steen
the literature. For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable
Counterexamples_in_Topology
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
or T3, and some authors allow a countable or even arbitrary number of isolated points. The existence of a Luzin space is independent of the axioms of
Luzin_space
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
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
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
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
Smallest ordinal number that, considered as a set, is uncountable
) {\displaystyle [0,\omega _{1})} is first-countable, but neither separable nor second-countable. The space [ 0 , ω 1 ] = ω 1 + 1 {\displaystyle [0,\omega
First_uncountable_ordinal
Topological space that is maximally disconnected
to a subset of a countable product of discrete spaces. It is in general not true that every open set in a totally disconnected space is also closed. It
Totally_disconnected_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
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
Normed vector space that is complete
theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with
Banach_space
Topological vector space whose topology can be defined by a metric
but at most countably many of these TVSs have the trivial topology. Every complete pseudometrizable TVS is a barrelled space and a Baire space (and thus
Metrizable topological vector space
Metrizable_topological_vector_space
Generalization of "n-th" to infinite cases
first uncountable cardinality. Cantor's second theorem becomes: If P′ is countable, then there is a countable ordinal α such that P(α) = ∅. Its proof
Ordinal_number
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)
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
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
Mathematical function revertible near each point
between two Hausdorff second-countable spaces where X {\displaystyle X} is a Baire space and Y {\displaystyle Y} is a normal space. If every fiber of f
Local_homeomorphism
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)
zero-dimensional compact Hausdorff space. It is a linearly ordered topological space that is separable but not second countable, hence not metrizable; its metrizable
Split_interval
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
Analog of Fubini's theorem for arbitrary second countable Baire spaces
theorem for arbitrary second countable Baire spaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces), and let A ⊂ X × Y
Kuratowski–Ulam_theorem
Mathematical concept
also second-countable (there are only finitely many open sets) and separable (since the space itself is countable). If a finite topological space is T1
Finite_topological_space
II. It is normal Haudsdorff, compact, separable, and first-countable but not second-countable. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology
Helly_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)
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
Example of a topology on the set of positive integers
subset, in a countable T1 space is a Gδ, so X is perfectly normal. X is countable, but not first countable, and hence not second countable and not metrizable
Appert_topology
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
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
Mathematical space with a notion of distance
original topological space (a disjoint union of countably many intervals) lead to different topologies on the quotient. A topological space is sequential if
Metric_space
Basic object in measure theory; set and a sigma-algebra
\left(X,{\mathcal {F}}_{2}\right).} If X {\displaystyle X} is finite or countably infinite, the σ {\displaystyle \sigma } -algebra is most often the power
Measurable_space
Topology on the real numbers
Hausdorff space. In terms of countability axioms, R l {\displaystyle \mathbb {R} _{l}} is first-countable and separable, but not second-countable. In terms
Lower_limit_topology
Branch of mathematical logic
express the principle "Every countable vector space has a basis" but it cannot express the principle "Every vector space has a basis". In practical terms
Reverse_mathematics
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 vector spaces
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. A locally convex quasibarrelled space that is also a σ-barrelled space is
Infrabarrelled_space
Generalization of the notion of convergence that is found in general topology
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain
Convergence_space
Rule in mathematics
{\displaystyle C(X)} has countable fan tightness. Compact space Sigma-compact Menger space Hurewicz space Rothberger space Menger, Karl (1924). "Einige
Selection_principle
Topological vector space
mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system
LF-space
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
Everything in space and time
responded to these paradoxes by developing the notion of a potential countable infinity, as well as the infinitely divisible continuum. The Indian philosopher
Universe
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
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
compact Hausdorff space that is second-countable, a similar argument can be used to construct an exhaustion. For a Hausdorff space X {\displaystyle X}
Exhaustion_by_compact_sets
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
Proof in set theory
that: The set T is uncountable. The proof starts by assuming that T is countable. Then all its elements can be written in an enumeration s1, s2, ... ,
Cantor's_diagonal_argument
Vector space with a notion of nearness
Fréchet spaces: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family
Topological_vector_space
Countable intersection of open sets
In general topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German
Gδ_set
Finite topological space with two points, only one of which is closed
homotopy groups). Like all finite topological spaces, the Sierpiński space is both compact and second-countable. The compact subset { 1 } {\displaystyle \{1\}}
Sierpiński_space
Schwartz space if the weak topology is Hausdorff. The locally convex strict inductive limit of any countable sequence of Schwartz spaces (with each space TVS-embedded
Schwartz topological vector space
Schwartz_topological_vector_space
In mathematics, vector space of linear forms
In mathematics, every vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms
Dual_space
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
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
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
condition that the space should also be a Fréchet space. (This means that the space is complete and the topology is given by a countable family of seminorms
Nuclear_space
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
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
Topology where the only open sets are the empty set and the entire space
topological space and codomain X is continuous. X is path-connected and so connected. X is second-countable, and therefore is first-countable, separable
Trivial_topology
X} is a DF-space, also written (DF)-space, if X {\displaystyle X} is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous
DF-space
Theorem in set-theoretic geometry
years earlier: the surface of a unit sphere in space is a disjoint union of three sets B, C, D and a countable set E such that, on the one hand, B, C, D are
Banach–Tarski_paradox
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SECOND COUNTABLE-SPACE
SECOND COUNTABLE-SPACE
SECOND COUNTABLE-SPACE
SECOND COUNTABLE-SPACE
SECOND COUNTABLE-SPACE
SECOND COUNTABLE-SPACE
SECOND COUNTABLE-SPACE
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