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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 {\displaystyle
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)
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
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
x} , and similarly, a filter subbase of the neighborhood filter of x {\displaystyle x} is known as a neighborhood subbase for x {\displaystyle x} . The
Filter_on_a_set
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
Maximal proper filter
\subseteq } ) filter containing a given filter subbase is said to be generated by the filter subbase. The upward closure in X {\displaystyle X} of a
Ultrafilter_on_a_set
Type of topology
axiom. If X is Hausdorff and S is a subbase for Y, then the collection {V(K, U) : U ∈ S, K compact} is a subbase for the compact-open topology on C(X
Compact-open_topology
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
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)
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
stabilized. If the final pavement is to have vehicle traffic, a layer of subbase of crushed stone or concrete must come next—this layer will even out the
Paver_base
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
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)
Use of filters to describe and characterize all basic topological notions and results
and filter subbases, all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/bases/subbases and uniformities
Filters_in_topology
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
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
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
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
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
Algebraic structure of set algebra
Filter Proper filter Never Never Never Prefilter (filter base) Filter subbase Open topology (even arbitrary ∪ {\displaystyle \cup } ) Never Closed topology
Σ-algebra
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
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
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
Finite ordered list of elements
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Tuple
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
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
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
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
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
One-to-one correspondence
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Bijection
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
Mathematician (1845–1918)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Georg_Cantor
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 } {\displaystyle \{x\mid a<x\}} { x ∣ x < b
Order_topology
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
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
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
Mathematical logician and philosopher
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Kurt_Gödel
especially analysts, use the term weaker topology. Subbase A collection of open sets is a subbase (or subbasis) for a topology if every non-empty proper
Glossary_of_general_topology
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)
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
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
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
Collection of mathematical objects
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Set_(mathematics)
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
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 set containing no elements
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Empty_set
Mathematical term
element of X ∗ {\displaystyle X^{*}} remains a continuous function. A subbase for the weak topology is the collection of sets of the form ϕ − 1 ( U )
Weak_topology
Lemma in constructibility theory
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Condensation_lemma
Land route
Layers in the construction of a mortarless pavement: A) Subgrade B) Subbase C) Base course D) Paver base E) Pavers F) Fine-grained sand
Road
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
Musical artist
Diesel Boy 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
Havensight and the Austin "Babe" Monsanto Marine Facility in Crown Bay (Subbase). Cruise ships can also anchor in the Charlotte Amalie Harbor and tender
Transportation in the United States Virgin Islands
Transportation_in_the_United_States_Virgin_Islands
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
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
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
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
English mathematician and philosopher (1872–1970)
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Bertrand_Russell
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
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
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
Technique invented by Paul Cohen for proving consistency and independence results
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Forcing_(mathematics)
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
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
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)
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
Locally convex topology on function spaces
The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U ( T 0 , x , ϵ ) = { T : ‖ T
Strong_operator_topology
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
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)
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
Product of any collection of compact topological spaces is compact
accumulation point. 2) The theorem is a quick corollary of the Alexander subbase theorem. More modern proofs have been motivated by the following considerations:
Tychonoff's_theorem
Branch of mathematics
principal ideals (i.e. sets of the form {y in X | y ≤ x} for some x) as a subbase. Additionally, a topology with specialization order ≤ may be order consistent
Order_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
Set theory concept
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Large_cardinal
Mathematical lemma
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Rasiowa–Sikorski_lemma
{\mathcal {B}}} will be a base for the topology. In general, it will be a subbase for the topology, and the set of all arithmetic progressions that are nonempty
Arithmetic progression topologies
Arithmetic_progression_topologies
Set of elements in any of some sets
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Union_(set_theory)
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
Generalization of compactness
{\displaystyle B_{j}} is U {\displaystyle U} -small. For any given filter subbase B {\displaystyle {\mathcal {B}}} of the identity element's neighborhood
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
Set that is not a finite set
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Infinite_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
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
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
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 the notion of convergence that is found in general topology
{\displaystyle \operatorname {Prefilters} (X),} FilterSubbases ( X ) , {\displaystyle \operatorname {FilterSubbases} (X),} UltraFilters ( X ) {\displaystyle
Convergence_space
Infinite set that is not countable
Set types Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset ·
Uncountable_set
Possibility of a consistent definition of "clockwise" in a mathematical space
{\displaystyle M} . To topologize O {\displaystyle O} we will specify a subbase for its topology. Let U {\displaystyle U} be an open subset of M {\displaystyle
Orientability
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
Mathematical proposition equivalent to the axiom of choice
more standard) proof of Tychonoff's theorem is to first prove Alexander's subbase lemma, but the proof of that lemma typically uses Zorn's lemma. There are
Zorn's_lemma
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
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
Coarse to fine grain rock materials used in concrete
other used materials.[citation needed] The material is crushed to provide subbase for roads and driveways, among other purposes. According to the USGS in
Construction_aggregate
Processes which convert waste plastic into new items
plastic's inclusion in asphalt concrete, (forming rubberised asphalt), subbase and recycled insulation can be beneficial. An example of this is the construction
Plastic_recycling
By-products of coal combustion
substitute material for clay) Mine reclamation Stabilization of soft soils Road subbase construction As aggregate substitute material (e.g. for brick production)
Coal_combustion_products
Element of interest in an algebraic structure
topology, a collection of sets that generate the topology is called a subbase Generating set of a topological algebra: S is a generating set of a topological
Generator_(mathematics)
SUBBASE
SUBBASE
SUBBASE
SUBBASE
Girl/Female
Bengali, Hindu, Indian
Victorious
Boy/Male
Indian
Bear
Boy/Male
Sikh
Temple of God
Girl/Female
French
Legend.
Female
Russian
(ЕвфимьÑ) Feminine form of Russian Efim, EFIM'IA means "holy, pious."Â
Girl/Female
Indian
Charitable, Good
Boy/Male
Tamil
Kesarisuta | கேஸரீஸà¯à®¤à®¾
(Son of Kesari)
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Marathi, Sanskrit, Tamil, Traditional
The Goddess of Wealth
Male
Hindi/Indian
(विशाल) Hindi name VISHAL means "large."
Girl/Female
Indian, Tamil
Always Ecstatic; Being Lucky
SUBBASE
SUBBASE
SUBBASE
SUBBASE
SUBBASE