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Topological space in which all singleton sets are closed
a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is
T1_space
Concept in topology
construct a T0 space by identifying topologically indistinguishable points. T0 spaces that are not T1 spaces are exactly those spaces for which the specialization
Kolmogorov_space
Type of topological space
normality of a space is purely a property of its lattice of open sets; cf. Ideal (order theory) § Prime and maximal spectra.) A T4 space is a T1 space X that
Normal_space
Topics referred to by the same term
Look up T1 in Wiktionary, the free dictionary. T1, T01, T.1 or T-1 may refer to: Target One, a Mesoamerican archaeological site in Honduras Tour T1, an office
T1
Topological space whose topology is fully captured by its lattice of open sets
sober space that is not T1 is the Sierpinski space. Moreover, T2 is strictly stronger than T1 and sober, i.e., while every T2 space is at once T1 and sober
Sober_space
Chinese lunar probe (2014–2020)
Chang'e 5-T1 (嫦娥五号T1; Cháng'é wǔhào T1) was an preparatory robotic lunar mission launched by China National Space Administration on 23 October 2014. It
Chang'e_5-T1
Type of topological space
separation axiom (after T0 and T1), which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used. A related, but weaker
Hausdorff_space
Theorem in topology
regular topological space (in fact, for a T1-space) to be paracompact. A family E i {\displaystyle E_{i}} of subsets of a topological space is said to be closure-preserving
Michael's theorem on paracompact spaces
Michael's_theorem_on_paracompact_spaces
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense
Luzin_space
Topological space where every open cover has a finite subcover
image of a supercompact space", Houston Journal of Mathematics, 5 (2): 241–247 Mysior, Adam (1992), "Universal compact T1-spaces", Canadian Mathematical
Supercompact_space
Battery electric compact hatchback
Beta T1 (Chinese: 极狐贝塔T1) is a battery electric compact hatchback manufactured by BAIC under the Arcfox brand. Initially sold as the Arcfox T1 in 2025
Arcfox_T1
Property of topological spaces stronger than normality
pairwise disjoint. Some authors assume that X {\displaystyle X} is also a T1 space as part of the definition, but no such assumption is made here. The property
Collectionwise_normal_space
Crash of rocket into the Moon
5-T1 mission on the far side of the Moon. Whilst this event posed no danger to satellites and other space assets, the impact renewed attention to space
2026_SpaceX_lunar_impact
Set of all limit points of a set
{\displaystyle S\subseteq X} if and only if it is a TD space. Every TD space is a T0 space. Every T1 space is a TD space, since every singleton is closed, hence {
Derived_set_(mathematics)
a discrete space with the subspace topology (i.e., all points of S {\displaystyle S} are isolated in S {\displaystyle S} ). Every T1 space that is collectionwise
Collectionwise Hausdorff space
Collectionwise_Hausdorff_space
Topological space whose topology has a countable base
second-countable space is second-countable, although uncountable products need not be. The topology of a second-countable T1 space has cardinality less
Second-countable_space
Space such that every point has a Hausdorff neighborhood
And in a locally Hausdorff space each point belongs to some Hausdorff dense open subspace. Every locally Hausdorff space is T1. The converse is not true
Locally_Hausdorff_space
Property of topological spaces
non trivial intersection. Hence, no T1 space with more than one point is ultraconnected. Every ultraconnected space X {\displaystyle X} is path-connected
Ultraconnected_space
Axioms in topology defining notions of "separation"
such things as "T1 space", "Fréchet topology", and "suppose that the topological space X is Fréchet"; one should avoid saying "Fréchet space" in this context
Separation_axiom
space, is scattered. This is an example of a scattered space that is not a T1 space. The closure of a scattered set is not necessarily scattered. For example
Scattered_space
Mathematical construction
point x {\displaystyle x} if the topological space in question is a T1 space, since every point of a T1 space is closed. This feature is the basis of the
Stalk_(sheaf)
Point not touching any other point
topological space is a point whose singleton is closed. In many areas of geometry and topology, all spaces under consideration are T1 spaces that only have
Closed_point
Topological space in which every point has a finite neighborhood
in F {\displaystyle {\mathcal {F}}} . A locally finite space is an Alexandrov space. A T1 space is locally finite if and only if it is discrete. Munkres
Locally_finite_space
of compactly generated weak Hausdorff spaces. Their strictness as separation properties in increasing order is T1: every single-point set is closed. Δ-Hausdorff:
Weak_Hausdorff_space
and variety of topological spaces. Dowker showed, in 1951, the following: If X is a normal T1 space (that is, a T4 space), then the following are equivalent:
Dowker_space
American mobile virtual network operator
Trump Mobile is an American mobile virtual network operator (MVNO) owned by T1 Mobile that uses a licensed brand from The Trump Organization. It was founded
Trump_Mobile
Topological space construction
a T1 space if and only if every equivalence class of ∼ {\displaystyle \sim } is closed in X . {\displaystyle X.} As an example, consider the space X =
Quotient_space_(topology)
Hence X {\displaystyle X} is not a Kolmogorov space, nor a T1 space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every
Partition_topology
Mathematical function
seminormable and T1 (because a TVS is Hausdorff if and only if it is a T1 space). A locally bounded topological vector space is a topological vector space that possesses
Seminorm
normal Hausdorff space (or T5 space) is a completely normal T1 space. (A completely normal space is Hausdorff if and only if it is T1, so the terminology
Glossary_of_general_topology
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
Countable union of closed sets
{\displaystyle \mathbb {R} } . More generally, any countable set in a T1 space is an Fσ set, because every singleton { x } {\displaystyle \{x\}} is closed
Fσ_set
Set of all points in a function's domain that all map to some single given point
(or more generally, the image set f ( X ) {\displaystyle f(X)} ) is a T1 space then every fiber is a closed subset of X . {\displaystyle X.} In particular
Fiber_(mathematics)
Topics referred to by the same term
descriptive set theory and general topology, Luzin space or Luzin set, a hypothetical uncountable topological T1 space without isolated points in which every nowhere-dense
Lusin_space
locally normal T1 space is locally regular and locally Hausdorff. A locally compact Hausdorff space is always locally normal. A normal space is always locally
Locally_normal_space
Characterization of normable spaces
Kolmogorov's normability criterion—A topological vector space is normable if and only if it is a T1 space and admits a bounded convex neighbourhood of the origin
Kolmogorov's normability criterion
Kolmogorov's_normability_criterion
"Small" subset of a topological space
(1,2]} is also nonmeagre. A countable T1 space without isolated point is meagre. So it is also meagre in any space that contains it as a subspace. For example
Meagre_set
topological space ( X , τ ) {\displaystyle (X,\tau )} is saturated if it is equal to an intersection of open subsets of X . {\displaystyle X.} In a T1 space every
Saturated_set
partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest. The specialization
Specialization_preorder
Topological space which is a generalization of certain compact spaces
and the sum of the nonzero functions is identically 1 in V. In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity
Paracompact_space
topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general)
Cosmic_space
discrete. A Hausdorff space cannot have a locally discrete basis unless it is itself discrete. The same property holds for a T1 space. The following is known
Locally_discrete_collection
Topological space that is maximally disconnected
and coproducts of totally disconnected spaces are totally disconnected. Totally disconnected spaces are T1 spaces, since connected components are closed
Totally_disconnected_space
for accessible spaces, Hausdorff spaces, regular spaces, and normal spaces. Topologists assigned these classes of spaces the names T1, T2, T3, and T4
History of the separation axioms
History_of_the_separation_axioms
Mathematical term
disjoint union of discrete spaces is discrete Separation Every disjoint union of T0 spaces is T0 Every disjoint union of T1 spaces is T1 Every disjoint union
Disjoint_union_(topology)
Generalization of manifolds
however a T1 space. The space is second countable. The space exhibits several phenomena that do not happen in Hausdorff spaces: The space is path connected
Non-Hausdorff_manifold
Vector space with a notion of nearness
X {\displaystyle X} to be T1; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is said to be separated if
Topological_vector_space
Axioms for defining a topology
(\{x\})=\{x\}} . He refers to topological spaces which satisfy all five axioms as T1-spaces in contrast to the more general spaces which only satisfy the four listed
Kuratowski_closure_axioms
Motor vehicle
The Caparo T1 is a British mid-engine, rear-wheel drive, two-seat automobile that was built by Caparo Vehicle Technologies, founded by design director
Caparo_T1
Satellite of Taiwan
to rocket failure. PARUS-T1 achieved orbit following a January 2025 launch as part of SpaceX's Transporter-12 mission. PARUS-T1 orbits at 515-kilometers
PARUS (Taiwanese satellite family)
PARUS_(Taiwanese_satellite_family)
Property of topological space
Niemytzki plane. In a first countable T1 space, every singleton is a Gδ set. That is not enough for the space to be a Gδ space, as shown for example by the lexicographic
Gδ_space
Branch of topology
may say such things as T1 space, Fréchet topology, and Suppose that the topological space X is Fréchet, avoid saying Fréchet space in this context, since
General_topology
compact space is limit point compact. For T1 spaces, countable compactness and limit point compactness are equivalent. Every sequentially compact space is
Countably_compact_space
{\displaystyle X} is X {\displaystyle X} itself. X {\displaystyle X} is a T1 space. A subset S {\displaystyle S} of a preordered set ( X , ≲ ) {\displaystyle
Saturated set (intersection of open sets)
Saturated_set_(intersection_of_open_sets)
Countable intersection of open sets
in pseudometrizable spaces. In a first countable T1 space, every singleton is a Gδ set. A subspace of a completely metrizable space X {\displaystyle X}
Gδ_set
space, together with the integer broom topology, is a compact topological space. It is a T0 space, but it is neither a T1 space nor a Hausdorff space
Integer_broom_topology
1]} with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlapping interval topology is second countable, with
Overlapping_interval_topology
Separable space Lindelöf space Sigma-compact space Connected space Simply connected space Path connected space T0 space T1 space Hausdorff space Completely
List of general topology topics
List_of_general_topology_topics
Carrier system for digital transmission of multiplexed telephone calls
multiplexed telephone calls. The first version, the Transmission System 1 (T1), was introduced in 1962 in the Bell System, and could transmit up to 24 telephone
T-carrier
Concept in mathematics
Indeed, in the category of T1 spaces and continuous maps, every object has a unique largest essential extension, but no space with more than one element
Essential_extension
Topological subset with no isolated point
dense-in-itself space, they include all open sets. In a dense-in-itself T1 space they include all dense sets. However, spaces that are not T1 may have dense
Dense-in-itself
for and engage in activities relating to outer space, including the operation of space systems, and space exploration.[citation needed] Their objectives
List of government space agencies
List_of_government_space_agencies
Property of topological spaces stronger than normality
satisfies any of the following equivalent definitions: The space X {\displaystyle X} is T1 and there is a function G {\displaystyle G} that assigns to
Monotonically_normal_space
Mathematical set whose closure has empty interior
empty set is nowhere dense. In a discrete space, the empty set is the only nowhere dense set. In a T1 space, any singleton set that is not an isolated
Nowhere_dense_set
Mathematical property of a space
the open set.) Equivalently, a space is T1 if all its singletons are closed. T1 spaces are always T0. Sober. A space is sober if every irreducible closed
Topological_property
Unmanned robotic spacecraft
A space probe is an uncrewed robotic spacecraft designed to explore outer space and transmit scientific data back to Earth. Space probes are used to investigate
Space_probe
does not contain y {\displaystyle y} . X {\displaystyle X} is a not a T1 space, as no point is closed. Consequently, X {\displaystyle X} is not Hausdorff
Divisor_topology
Esports series
Sports Park Multifunctional Gymnasium in Chengdu, China between KT Rolster and T1. It marked the fifteenth final of a LoL World Championship and the first championship
2025 League of Legends World Championship final
2025_League_of_Legends_World_Championship_final
Office skyscraper in La Défense, the high-rise business district west of Paris, France
of the building's office space has been rented by GDF Suez. Skyscraper La Défense List of tallest structures in Paris "Tour T1". Skyscraper Center. CTBUH
Tour_T1
Group obtained by aggregating similar elements of a larger group
Indeed, if N {\displaystyle N} is not closed then the quotient space is not a T1-space (since there is a coset in the quotient which cannot be separated
Quotient_group
Characterization of normal spaces by continuous functions
'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal T1 spaces are Tychonoff. A topological space X {\displaystyle X} is
Urysohn's_lemma
Set of a ring's prime ideals
(R)} is not, in general, a T1 space. However, Spec ( R ) {\displaystyle \operatorname {Spec} (R)} is always a Kolmogorov space (satisfies the T0 axiom);
Spectrum_of_a_ring
Concept category theory (mathematics)
(\{0\leftrightarrow 1\}\to \{*\})^{\perp r}} .[clarification needed] A space X {\displaystyle X} is a T1 space if and only if ∅ → X {\displaystyle \emptyset \to X} is
Lifting_property
Example of a topology on the set of positive integers
Finally, any subset, in particular any closed subset, in a countable T1 space is a Gδ, so X is perfectly normal. X is countable, but not first countable
Appert_topology
German optics company
7 mm T1.6 DigiPrime T✻ 10 mm T1.6 DigiPrime T✻ 14 mm T1.6 DigiPrime T✻ 20 mm T1.6 DigiPrime T✻ 28 mm T1.6 DigiPrime T✻ 40 mm T1.6 DigiPrime T✻ 52 mm T1.6
Zeiss_(company)
Series of vans
(Volkswagen Beetle), the Volkswagen Type 2 (T1) was the first generation of Volkswagen's Transporter family. T1 (front) T1 (rear) The Volkswagen T2 platform was
Volkswagen_Transporter
Mathematical space with a notion of closeness
In mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance
Topological_space
topological space could be defined in terms of its distributive lattice of closed sets. He observed that the inclusion order on the closed sets of a T1 space has
Disjunction property of Wallman
Disjunction_property_of_Wallman
Space homeomorphic to some ring spectrum
but in general not T1. In fact a spectral space is T1 if and only if it is Hausdorff (i.e. T2) if and only if it is a boolean space if and only if K ∘
Spectral_space
Type of topological space in mathematics
uncountable set. Every countably compact space (and hence every compact space) is limit point compact. For T1 spaces, limit point compactness is equivalent
Limit_point_compact
NASA-led lunar exploration program
exploration program led by the United States' National Aeronautics and Space Administration (NASA), aimed at returning humans to the Moon for the first
Artemis_program
Volkswagen panel van
flat-four engine in the rear, and all riding on the same (initial thirty years—T1 and T2), or similar (T3), 2.40 m (94 in) wheelbase as the Type 1 Beetle. As
Volkswagen_Type_2
Complement of an open subset
points). Singleton points (and thus finite sets) are closed in T1 spaces and Hausdorff spaces. The set of integers Z {\displaystyle \mathbb {Z} } is an infinite
Closed_set
Topology on Cartesian products of topological spaces
product space X {\displaystyle X} . Separation Every product of T0 spaces is T0. Every product of T1 spaces is T1. Every product of Hausdorff spaces is Hausdorff
Product_topology
Maximally symmetric Lorentzian manifold with a negative cosmological constant
space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. It is the Lorentzian analogue of hyperbolic space.
Anti-de_Sitter_space
for a T1 space being fully normal and being paracompact are equivalent. This was a landmark theorem and provided the first proof that metric spaces are
Star_refinement
Mathematical generalization of boundedness
{\displaystyle X} in which singleton subsets are relatively compact (such as a T1 space), the set of all relatively compact subsets of X {\displaystyle X} form
Bornology
Decay of nuclear spin polarization in MRI and NMR
conventional NMR spectroscopy, T1 limits the pulse repetition rate and affects the overall time an NMR spectrum can be acquired. Values of T1 range from milliseconds
Relaxation_(NMR)
jet training operating the Beechcraft Texan T1 and BAE Systems Hawk T2 Defence College of Air and Space Operations – air traffic control and battlespace
Structure of the Royal Air Force
Structure_of_the_Royal_Air_Force
China has one of the most active space programs in the world. With launch vehicles of the Long March rocket family and four spaceports (Jiuquan, Taiyuan
Chinese_space_program
Examples of topological spaces
all but a finite number of points of X. The space X is compact and T1, but not Hausdorff. Fortissimo space is defined by taking an uncountable set X, with
Fort_space
Renewed 21st-century competition in space exploration
The New Space Race or Second Space Race describes the renewed competition in space exploration and technological development in the 21st century, primarily
New_Space_Race
List of deliberate crash landings on extraterrestrial bodies
provenance; no space program has taken credit for it, although a later study attributed it to a spent upper stage from the Chang'e 5-T1 mission. The Deep
List of spacecraft intentionally crashed into extraterrestrial bodies
List_of_spacecraft_intentionally_crashed_into_extraterrestrial_bodies
American digital cinematography company
f2.8 lens was discontinued early. Other lenses like a RED Mini Prime 6 mm T1.5 and a RED Mini Prime 50mm T2.9 with coverage for 2/3" inch sensors were
Red_Digital_Cinema
Turkish aerospace and defense company
lifecycle support of the aviation and space industry systems. Being among the top hundred global players in aviation and space industry, Turkish Aerospace is
Turkish_Aerospace_Industries
Variants of the Eurofighter Typhoon
Unit (OCU).[citation needed] Typhoon T1 The Typhoon T1 is a Tranche 1, batch 1 two-seat trainer. The first Typhoon T1 is one of the Instrumented Production
Eurofighter_Typhoon_variants
Motor vehicle
Park Ward. T1 interior 1970 T1 Four door saloon 1967 T1, Mulliner Park Ward two-door saloon (on later wheels) Pininfarina's one-off 1968 T1 Coupe Speciale
Bentley_T-series
Supertall skyscraper in Changsha, Hunan, China
June 6, 2021. "Changsha IFS Tower T1 Fact Sheet". buildingdb.ctbuh.org. Retrieved June 6, 2021. "Changsha IFS Tower T1". The Skyscraper centre. Council
Changsha_IFS_Tower
Air and space warfare force of the United Kingdom
Operational Conversion Unit or front-line squadron. Viking T1 Tutor T1 Prefect T1 Texan T1 Hawk T2 Phenom T1 Typhoon T3 No. 1 Flying Training School (No. 1 FTS)
Royal_Air_Force
Medical imaging technique
equilibrium state after excitation by the independent relaxation processes of T1 (spin-lattice; that is, magnetization in the same direction as the static
Magnetic_resonance_imaging
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