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Collection of subsets that generate a topology
In topology in mathematics, a subbase (or subbasis, prebase, prebasis) for the topology τ of a topological space (X, τ) is a subcollection B of τ that
Subbase
Layer of a paved road
In highway engineering, subbase is the layer of aggregate material laid on the subgrade, on which the base course layer is located. It may be omitted
Subbase_(pavement)
Theorem in topology
In mathematics, specifically topology, Alexander's subbase lemma states that for a topological space to be a compact, it is necessary and sufficient that
Alexander's_subbase_lemma
Composite material used for paving
The pavement and subbase thicknesses are designed to withstand the wheel loads. Sometimes, geogrids are used to reinforce the subbase and further strengthen
Asphalt_concrete
Family of subsets representing "large" sets
neighborhood base for x, and similarly, a filter subbase of the neighborhood filter of x is known as a neighborhood subbase for x. The open neighborhoods of x always
Filter_on_a_set
Maximal proper filter
(relative to ⊆) filter containing a given filter subbase is said to be generated by the filter subbase. The upward closure in X of a family of sets P is
Ultrafilter_on_a_set
Civil engineering of roads, bridges, and tunnels
course, and subbase course constructed over a compacted, natural soil subgrade. When building a three layer flexible pavement, the subbase layer is not
Highway_engineering
Collection of open sets used to define a topology
used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood
Base_(topology)
Groove in a road caused by vehicle traffic
can form through the deformation of the asphalt concrete, pavement or subbase material. In modern roads the main cause is heavily loaded trucks. These
Rut_(roads)
Concept in mathematics
convex sets Neighbourhood (mathematics) – Open set containing a given point Subbase – Collection of subsets that generate a topology Tubular neighborhood –
Neighbourhood_system
Distress in asphalt pavement
the direct cause. Frequently, overloading happens because the base or subbase inadequately support the surface layer and subsequently cannot handle loads
Crocodile_cracking
Upper layer in roadway, airfield, and dockyard construction
which is then laid on the base course, which is normally placed on the subbase, which rests on the subgrade. There are various different types of flexible
Wearing_course
Material underneath a road or track
English it is called formation level. The subgrade provides support to the subbase level and acts as an integral load-bearing layer. Failure of the subgrade
Subgrade
Space with topology generated by convex sets
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological
Locally convex topological vector space
Locally_convex_topological_vector_space
bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10
List_of_numeral_systems
Type of topology
V(K,U)=C(K,U)×C(K,Y)C(X,Y). Then the collection of all such V(K, U) is a subbase for the compact-open topology on C(X, Y). (This collection generally does
Compact-open_topology
Road covered with durable surface material
open ditch or covered drains to reduce moisture content in the pavements subbase and subgrade. Wheeled transport created the need for better roads. Generally
Road_surface
Property in general topology
finite intersection property are also called centered systems and filter subbases. The finite intersection property can be used to reformulate topological
Finite_intersection_property
Collective term for all forms of transport which takes place on roads
vehicle loads and faster speeds, requiring thicker slabs and deeper subbase. Subbase is the layer or successive layers of stone, gravel and sand supporting
Road_transport
Algebraic structure of set algebra
Filter Proper filter Never Never Never Prefilter (filter base) Filter subbase Open topology (even arbitrary ∪) Never Closed topology (even arbitrary
Σ-algebra
US Air Force base near Mountain Home, Idaho, United States
Temporary inactive status, 5 October 1945. Subbase of Gowen Army Airfield (Boise), Idaho, 9 October 1945 Subbase of Walla Walla Army Airfield, Washington
Mountain_Home_Air_Force_Base
Special subset of a partially ordered set
set U of all finite intersections of T. The set T is said to be a filter subbase when F (and thus U) is proper. Proper filters on sets have the finite intersection
Filter_(mathematics)
Layer of material in asphalt roadways
Layers in the construction of a mortarless pavement: A. Subgrade B. Subbase C. Base course D. Paver base as binder course E. Pavers as wearing course
Base_course
Certain topology in mathematics
is a totally ordered set, the order topology on X is generated by the subbase of "open rays" {x∣a<x} {x∣x<b} for all a, b in X. Provided X has at least
Order_topology
Stone or tile structure which can serve as floor; pavement type with solid blocks
Layers in the construction of a mortarless pavement: A.) Subgrade B.) Subbase C.) Base course D.) Paver base E.) Pavers F.) Fine-grained sand.
Pavers_(flooring)
Standard system of axiomatic set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Zermelo–Fraenkel_set_theory
Finite ordered list of elements
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Tuple
Set whose elements all belong to another set
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Subset
Diagram that shows all possible logical relations between a collection of sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Venn_diagram
Mathematical concept
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Equivalence_class
mining management can include: number of lanes road width road slope road subbase material geometry of curves of road Determination of the haul trucks gross
Haul_road
Mathematical generalization of boundedness
B\subseteq A.} A family of sets S {\displaystyle {\mathcal {S}}} is called a subbase of a bornology B {\displaystyle {\mathcal {B}}} if S ⊆ B {\displaystyle
Bornology
that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology. Complete A metric space is complete if every
Glossary_of_general_topology
Cellular confinement system
confinement system is used to stabilize soft subgrade soil and reinforce the subbase and base layers in flexible pavements. Cellular confinement is also used
Neoloy_Geocell
Branch of mathematics that studies sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Set_theory
German-Israeli mathematician and Zionist (1891–1965)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Abraham_Fraenkel
Department of Transport and Planning/VicRoads (2020) - Section 813: Base and subbase for lower trafficked roads. VicRoads (2019) - Technical Note 107: Use of
Arul_Arulrajah
Hungarian and American mathematician and physicist (1903–1957)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
John_von_Neumann
One-to-one correspondence
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Bijection
Construction material with soil and portland cement
material for pipe bedding, slope protection, and road construction as a subbase layer reinforcing and protecting the subgrade. It has good compressive
Soil_cement
Pair of logical equivalences
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
De_Morgan's_laws
Method to utilize rock debris from tunneling
into other usable needs. The most common is for concrete aggregates or as subbase for road building. Crushers and screeners normally used in quarries are
Tunnel_rock_recycling
Language of ancient Sumer and Babylon
so that the Sumerian lexical numeral system is sexagesimal with 10 as a subbase. The cardinal numerals and ways of forming composite numbers are as follows:
Sumerian_language
Mathematical set of all subsets of a set
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Power_set
Paradox in set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Russell's_paradox
Mathematician and philosopher (1906–1978)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Kurt_Gödel
Set of the elements not in a given subset
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Complement_(set_theory)
German mathematician (1831–1916)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Richard_Dedekind
Technique for proving independence results
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Forcing_(mathematics)
Collection of mathematical objects
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Set_(mathematics)
Family of sets closed under countable unions
Filter Proper filter Never Never Never Prefilter (filter base) Filter subbase Open topology (even arbitrary ∪) Never Closed topology (even arbitrary
Sigma-ring
Set of elements common to all of some sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Intersection_(set_theory)
Axiom of set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_of_choice
Czech mathematician
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Thomas_Jech
Ring closed under countable intersections
Filter Proper filter Never Never Never Prefilter (filter base) Filter subbase Open topology (even arbitrary ∪) Never Closed topology (even arbitrary
Delta-ring
Concept in mathematics
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_of_countable_choice
Proof in set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Cantor's_diagonal_argument
Confinement system used in construction and geotechnical engineering
thickness. Typical load support applications include reinforcement of base and subbase layers in flexible pavements, including: asphalt pavements; unpaved access
Cellular_confinement
Mathematician (1845–1918)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Georg_Cantor
Axiom of Zermelo-Fraenkel set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_of_infinity
Mathematical set containing no elements
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Empty_set
Term in set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Almost
Type of numeral system
expressing integers. With this sexagesimal positional system – with a subbase of 10 – for expressing fractions, fourteen of the alphabetic numerals were
Alphabetic_numeral_system
Topological space in which all singleton sets are closed
definition of O A {\displaystyle O_{A}} from the previous example, define a subbase of open sets G x {\displaystyle G_{x}} for any integer x {\displaystyle
T1_space
Paradox in set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Burali-Forti_paradox
Elements in exactly one of two sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Symmetric_difference
British record label
History Of Hardcore (1995) (collaboration with Moving Shadow) Subbase Sampler (1996) Classic Subbase (1997) Suburban Base Records (The History of Hardcore, Jungle
Suburban_Base
Size of a possibly infinite set
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Cardinal_number
Concept in set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_schema_of_replacement
Possible axiom for set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_of_determinacy
Musical artist
Dieselboy and DJ Dara to create the annual Planet of the Drums tour. 1996 Subbase Classics 1998 Fully Automatic 1999 Lock & Roll: A Drum & Bass DJ Mix 2000:
AK1200
Rail infrastructure
high-performance geogrids at the subgrade and high-performance geocells in the upper subbase/subballast layer has been shown to increase the reinforcement factor beyond
Railway_track
System for writing in Spanish
connotar, dígannos), and ⟨b⟩ (in a few words with the prefix sub-; e.g. subbase, subbético). Exceptions to this limitation are gamma (and its derivatives
Spanish_orthography
Mathematical set formed from two given sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Cartesian_product
Mathematical set that can be enumerated
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Countable_set
Mathematical concept for comparing objects
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Equivalence_relation
System of mathematical set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Zermelo_set_theory
US Air Force base near Dover, Delaware
Pearl Harbor. It was renamed Dover Army Air Base on April 8, 1943; Dover Subbase on June 6, 1943, and Dover Army Airfield on February 2, 1944. With the
Dover_Air_Force_Base
Alternative to the standard Zermelo–Fraenkel set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
List of alternative set theories
List_of_alternative_set_theories
Topological space where every open cover has a finite subcover
superextension was introduced by J. de Groot in 1967. By the Alexander subbase theorem, every supercompact space is compact. Conversely, many (but not
Supercompact_space
Type of unpaved road surfaced with gravel
gravel road will have a higher percentage of fines than gravel used as a subbase for a paved road. This often causes problems if a gravel road is paved
Gravel_road
Concept in axiomatic set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_of_power_set
Proposition in mathematical logic
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Continuum_hypothesis
obtained by taking as a subbase the collection of all pℕ+b with b>0 and p prime not dividing b. Equivalently, one can take as a subbase the collection of all
Arithmetic progression topologies
Arithmetic_progression_topologies
Swiss mathematician (1888–1977)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Paul_Bernays
Set with exactly one element
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Singleton_(mathematics)
3-volume treatise on mathematics, 1910–1913
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Principia_Mathematica
Concept in axiomatic set theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Axiom_schema_of_specification
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
List_of_types_of_sets
Generalization of compactness
of X such that S⊆B1∪⋯∪Bn and each Bj is U-small. For any given filter subbase ℬ of the identity element's neighborhood filter 𝒩 (which consists of all
Totally_bounded_space
Finite sets whose elements are all hereditarily finite sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Hereditarily_finite_set
Identities and relationships involving sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Algebra_of_sets
Proof by Alan Turing
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Turing's_proof
German logician and mathematician (1871–1953)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Ernst_Zermelo
Unsolved problem in number theory
1956 result to bases of arbitrary order. In 2000, V. Vu proved that thin subbases exist in the Waring bases using the Hardy–Littlewood circle method and
Erdős–Turán conjecture on additive bases
Erdős–Turán_conjecture_on_additive_bases
Coarsest topology making certain functions continuous
The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology. If the space X {\displaystyle
Initial_topology
Finite collection of distinct objects
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Finite_set
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Paradoxes_of_set_theory
Generalization of "n-th" to infinite cases
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Ordinal_number
Equalities for combinations of sets
that is denoted by π(ℒ) and called the π−system generated by ℒ. filter subbase and is said to have the finite intersection property if ℒ≠∅ and ∅∉π(ℒ)
List of set identities and relations
List_of_set_identities_and_relations
travel, tourism, insurance
SUBBASE
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SUBBASE
travel, tourism, insurance