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See the article on Topological Space s for basic definitions and examples, and see the article on Topology for a brief history and description of the subject area. See Naive Set Theory , Axiomatic Set Theory , and Function for definitions concerning sets and functions. The following articles may also be useful. These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The List Of General Topology Topics and the List Of Examples In General Topology will also be very helpful.
All spaces in this glossary are assumed to be Topological Space s unless stated otherwise. __NOTOC__ A ;Accessible: See T1 . ;Accumulation point: See Limit Point . ; (or is finitely generated) if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed. ;Almost discrete: A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces. ; is a generalization of metric space based on point-to-set distances, instead of point-to-point. B ;Baire space: This has two distinct common meanings: :#A space is a Baire space if the intersection of any Countable collection of dense open sets is dense; see Baire Space . :#Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire Space (set Theory) . ; (or basis) for a topology ''T'' if every open set in ''T'' is a union of sets in ''B''. The topology ''T'' is the smallest topology on ''X'' containing ''B'' and is said to be generated by ''B''. ;'''. ; on a space is the smallest σ-algebra containing all the open sets. ;Borel set: A Borel set is an element of a Borel algebra. ; (or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. ; if it has Finite diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A Function taking values in a metric space is Bounded if its Image is a bounded set. C ; Top has Topological Space s as Objects and Continuous Map s as Morphism s. ; {''x''''n''} in a metric space (''M'', ''d'') is a Cauchy Sequence if, for every Positive Real Number ''r'', there is an Integer ''N'' such that for all integers ''m'', ''n'' > ''N'', we have ''d''(''x''''m'', ''x''''n'') < ''r''. ; if it is both open and closed. ;Closed ball: If (''M'', ''d'') is a Real Number , the radius of the ball. A closed ball of radius ''r'' is a '''closed ''r''-ball'''. Every closed ball is a closed set in the topology induced on ''M'' by ''d''. Note that the closed ball ''D''(''x''; ''r'') might not be equal to the Closure of the open ball ''B''(''x''; ''r''). ; if its complement is a member of the topology. ;Closed function: A function from one space to another is closed if the Image of every closed set is closed. ; of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set ''S'' is a point of closure of ''S''. ;Closure operator: See Kuratowski Closure Axioms . ; (or smaller, '''weaker''') than ''T''2 if ''T''1 is contained in ''T''2. Beware, some authors, especially Analyst s, use the term '''stronger'''. ;Comeagre: A subset ''A'' of a space ''X'' is comeagre ('''comeager''') if its Complement ''X''\''A'' is Meagre . Also called '''residual'''. ; if every open cover has a Finite subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff Space is normal. See also quasicompact. ; on the set ''C''(''X'', ''Y'') of all continuous maps between two spaces ''X'' and ''Y'' is defined as follows: given a compact subset ''K'' of ''X'' and an open subset ''U'' of ''Y'', let ''V''(''K'', ''U'') denote the set of all maps ''f'' in ''C''(''X'', ''Y'') such that ''f''(''K'') is contained in ''U''. Then the collection of all such ''V''(''K'', ''U'') is a subbase for the compact-open topology. ; if every Cauchy sequence converges. ;Completely metrizable/completely metrisable: See Complete Space . ;Completely normal: A space is completely normal if any two separated sets have Disjoint neighbourhoods. ;Completely normal Hausdorff: A completely 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 is consistent.) Every completely normal Hausdorff space is normal Hausdorff. ; if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and {''x''} are functionally separated. ;'''. ;Component: See Connected Component /'''Path-connected component'''. ; if it is not the union of a pair of Disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set. ; of a space is a Maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a Partition of that space. ; if the Preimage of every open set is open. ; on ''X'' is homotopic to a Constant map. Every contractible space is simply connected. ;Coproduct topology: If {''X''''i''} is a collection of spaces and ''X'' is the (set-theoretic) Disjoint Union of {''X''''i''}, then the coproduct topology (or disjoint union topology, '''topological sum''' of the ''X''''i'') on ''X'' is the finest topology for which all the injection maps are continuous. ;Countably compact: A space is countably compact if every Countable open cover has a Finite subcover. Every countably compact space is pseudocompact and weakly countably compact. ;Countably locally finite: A collection of subsets of a space ''X'' is countably locally finite (or '''σ-locally finite''') if it is the union of a Countable collection of locally finite collections of subsets of ''X''. ; Cover : A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is the whole space. ;Covering: See Cover. ;Cut point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a cut point if the subspace ''X'' − {''x''} is disconnected. D ; Dense Set : A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space. ;Derived set: If ''X'' is a space and ''S'' is a subset of ''X'', the derived set of ''S'' in ''X'' is the set of limit points of ''S'' in ''X''. ;Diameter: If (''M'', ''d'') is a metric space and ''S'' is a subset of ''M'', the diameter of ''S'' is the Supremum of the distances ''d''(''x'', ''y''), where ''x'' and ''y'' range over ''S''. ;Discrete metric: The discrete metric on a set ''X'' is the function ''d'' : ''X'' × ''X'' → R such that for all ''x'', ''y'' in ''X'', ''d''(''x'', ''x'') = 0 and ''d''(''x'', ''y'') = 1 if ''x'' ≠ ''y''. The discrete metric induces the discrete topology on ''X''. ; if every subset of ''X'' is open. We say that ''X'' carries the discrete topology. ;'''. ;Disjoint union topology: See Coproduct topology. ;Dispersion point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a dispersion point if the subspace ''X'' − {''x''} is totally disconnected. ;Distance: See Metric Space . ; Dunce Hat (topology) E ;'''. ;Exterior: The exterior of a set is the interior of its complement. F ; is a Countable union of closed sets. ;Filter: A filter on a space ''X'' is a nonempty family ''F'' of subsets of ''X'' such that the following conditions hold: :# The Empty Set is not in ''F''. :# The intersection of any Finite number of elements of ''F'' is again in ''F''. :# If ''A'' is in ''F'' and if ''B'' contains ''A'', then ''B'' is in ''F''. ; (or larger, '''stronger''') than ''T''1 if ''T''2 contains ''T''1. Beware, some authors, especially Analyst s, use the term '''weaker'''. ;Finitely generated: See Alexandrov Topology . ;'''. ; if every point has a Countable local base. ;Fréchet: See T1. ;Frontier: See Boundary . ;Functionally separated: Two sets ''A'' and ''B'' in a space ''X'' are functionally separated if there is a continuous map ''f'': ''X'' → 1 such that ''f''(''A'') = 0 and ''f''(''B'') = 1. G ; is a Countable intersection of open sets. H ; (or T2 space) is one in which every two distinct points have Disjoint neighbourhoods. Every Hausdorff space is T1. ;Hereditary: A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it. For example, second-countability is a hereditary property. ; from ''X'' to ''Y'' is a Bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''−1 are continuous. The spaces ''X'' and ''Y'' are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical. ; is homogeneous. ; (in ''Y'') if there is a continuous map ''H'' : ''X'' × 1 → ''Y'' such that ''H''(''x'', 0) = ''f''(''x'') and ''H''(''x'', 1) = ''g''(''x'') for all ''x'' in ''X''. Here, ''X'' × 1 is given the product topology. The function ''H'' is called a homotopy (in ''Y'') between ''f'' and ''g''. ;Homotopy: See Homotopic Maps . ; Hyper-connected : A space is hyper-connected if no two non-empty open sets are disjoint. Every hyper-connected space is connected. I ;Identification map: See Quotient map. ;'''. ;'''. ; of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set ''S'' is an interior point of ''S''. ;Interior point: See Interior . ; if the Singleton {''x''} is open. More generally, if ''S'' is a subset of a space ''X'', and if ''x'' is a point of ''S'', then ''x'' is an isolated point of ''S'' if {''x''} is open in the subspace topology on ''S''. ;Isometric isomorphism: If ''M''1 and ''M''2 are metric spaces, an isometric isomorphism from ''M''1 to ''M''2 is a Bijective isometry ''f'' : ''M''1 → ''M''2. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical. ;Isometry: If (''M''1, ''d''1) and (''M''2, ''d''2) are metric spaces, an isometry from ''M''1 to ''M''2 is a function ''f'' : ''M''1 → ''M''2 such that ''d''2(''f''(''x''), ''f''(''y'')) = ''d''1(''x'', ''y'') for all ''x'', ''y'' in ''M''1. Every isometry is Injective , although not every isometry is Surjective . K ;. ; is a set of Axiom s satisfied by the function which takes each subset of ''X'' to its closure: :# '' Isotonicity '': Every set is contained in its closure. :# '' Idempotence '': The closure of the closure of a set is equal to the closure of that set. :# ''Preservation of binary unions'': The closure of the union of two sets is the union of their closures. :# ''Preservation of nullary unions'': The closure of the empty set is empty. :If ''c'' is a function from the Power Set of ''X'' to itself, then ''c'' is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on ''X'' by declaring the closed sets to be the Fixed Point s of this operator, i.e. a set ''A'' is closed If And Only If ''c''(''A'') = ''A''. L ;Larger topology: See Finer Topology . ; of a subset ''S'' if every open set containing ''x'' also contains a point of ''S'' other than ''x'' itself. This is equivalent to requiring that every neighbourhood of ''x'' contains a point of ''S'' other than ''x'' itself. ;Limit point compact: See Weakly countably compact. ; if every open cover has a Countable subcover. ; Local Base : A set ''B'' of neighbourhoods of a point ''x'' of a space ''X'' is a local base (or local basis, '''neighbourhood base''', '''neighbourhood basis''') at ''x'' if every neighbourhood of ''x'' contains some member of ''B''. ;Local basis: See Local base. ;Locally closed subset: A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure. ; if every point has a local base consisting of compact neighbourhoods. Every locally compact Hausdorff space is Tychonoff. ; if every point has a local base consisting of connected neighbourhoods. ;Locally finite: A collection of subsets of a space is locally finite if every point has a neighbourhood which has nonempty intersection with only Finite ly many of the subsets. See also countably locally finite, ''' Point Finite '''. ;Locally metrizable/Locally metrisable: A space is locally metrizable if every point has a metrizable neighbourhood. ; if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected If And Only If it is path-connected. ;Locally simply connected: A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods. ; at ''x'' in ''X'' (or a loop in ''X'' with basepoint ''x'') is a path ''f'' in ''X'', such that ''f''(0) = ''f''(1) = ''x''. Equivalently, a loop in ''X'' is a continuous map from the Unit Circle ''S''1 into ''X''. M ; union of nowhere dense sets. If ''A'' is not meagre in ''X'', ''A'' is of second category in ''X''. ;Metric: See Metric Space . ;Metric invariant: A metric invariant is a property which is preserved under isometric isomorphism. ; satisfying the following axioms for all ''x'', ''y'', and ''z'' in ''M'': :# ''d''(''x'', ''y'') ≥ 0 :# ''d''(''x'', ''x'') = 0 :# if ''d''(''x'', ''y'') = 0 then ''x'' = ''y'' (''identity of indiscernibles'') :# ''d''(''x'', ''y'') = ''d''(''y'', ''x'') (''symmetry'') :# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') ('' Triangle Inequality '') :The function ''d'' is a metric on ''M'', and ''d''(''x'', ''y'') is the '''distance''' between ''x'' and ''y''. The collection of all open balls of ''M'' is a base for a topology on ''M''; this is the topology on ''M'' induced by ''d''. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable. ; if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable. ;Monolith: Every non-empty ultra-connected compact space ''X'' has a largest proper open subset; this subset is called a monolith. N ; set {''x''}. (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.) ;'''. ; Neighbourhood System ''' for a point ''x'': A neighbourhood system at a point ''x'' in a space is the collection of all neighbourhoods of ''x''. ; in a space ''X'' is a map from a Directed Set ''A'' to ''X''. A net from ''A'' to ''X'' is usually denoted (''x''α), where α is an Index Variable ranging over ''A''. Every Sequence is a net, taking ''A'' to be the directed set of Natural Number s with the usual ordering. ; if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a partition of unity. ; space (or T4 Space ) is a normal T1 space. (A normal space is Hausdorff If And Only If it is T1, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff. ; set is a set whose closure has empty interior. O ; is a cover consisting of open sets. ;Open ball: If (''M'', ''d'') is a metric space, an open ball is a set of the form ''B''(''x''; ''r'') := {''y'' in ''M'' : ''d''(''x'', ''y'') < ''r''}, where ''x'' is in ''M'' and ''r'' is a Positive Real Number , the radius of the ball. An open ball of radius ''r'' is an '''open ''r''-ball'''. Every open ball is an open set in the topology on ''M'' induced by ''d''. ;Open condition: See open property. ; is a member of the topology. ; if the Image of every open set is open. ;Open property: A property of points in a Topological Space is said to be "open" if those points which possess it form an Open Set . Such conditions often take a common form, and that form can be said to be an ''open condition''; for example, in Metric Space s, one defines an open ball as above, and says that "strict inequality is an open condition". P ; if every open cover has a locally finite open refinement. Paracompact Hausdorff spaces are normal. ; number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1. ; in a space ''X'' is a continuous map ''f'' from the closed unit Interval 1 into ''X''. The point ''f''(0) is the initial point of ''f''; the point ''f''(1) is the terminal point of ''f''. ; if, for every two points ''x'', ''y'' in ''X'', there is a path ''f'' from ''x'' to ''y'', i.e., a path with initial point ''f''(0) = ''x'' and terminal point ''f''(1) = ''y''. Every path-connected space is connected. ;Path-connected component: A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a Partition of that space, which is Finer than the partition into connected components. The set of path-connected components of a space ''X'' is denoted π0(''X'') . ;Point: A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point". ;Point of closure: See Closure . ; Polish : A space is Polish if it is separable and topologically complete, i.e. if it is homeomorphic to a separable and complete metric space. ;Pre-compact: See Relatively Compact . ; of {''X''''i''}, then the Product Topology on ''X'' is the coarsest topology for which all the projection maps are continuous. ;Proper function/mapping: A continuous function ''f'' from a space ''X'' to a space ''Y'' is proper if ''f''−1(''C'') is a compact set in ''X'' for any compact subspace ''C'' of ''Y''. ; δ between subsets of ''X'' satisfying the following properties: :For all subsets ''A'', ''B'' and ''C'' of ''X'', :#''A'' δ ''B'' implies ''B'' δ ''A'' :#''A'' δ ''B'' implies ''A'' is non-empty :#If ''A'' and ''B'' have non-empty intersection, then ''A'' δ ''B'' :#''A'' δ (''B'' ∪ ''C'') Iff (''A'' δ ''B'' or ''A'' δ ''C'') :#If, for all subsets ''E'' of ''X'', we have (''A'' δ ''E'' or ''B'' δ ''E''), then we must have ''A'' δ (''X'' − ''B'') ;Pseudocompact: A space is pseudocompact if every Real-valued continuous function on the space is bounded. ;Pseudometric: See Pseudometric space. ;Pseudometric space: A pseudometric space (''M'', ''d'') is a set ''M'' equipped with a function ''d'' : ''M'' × ''M'' → R satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. The function ''d'' is a '''pseudometric''' on ''M''. Every metric is a pseudometric. ;Punctured neighbourhood/Punctured neighborhood: A punctured neighbourhood of a point ''x'' is a neighbourhood of ''x'', minus {''x''}. For instance, the , so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0. Q ;Quasicompact: See Compact . Some authors define "compact" to include the Hausdorff separation axiom, and they use the term '''quasicompact''' to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French. ;Quotient map: If ''X'' and ''Y'' are spaces, and if ''f'' is a Surjection from ''X'' to ''Y'', then ''f'' is a quotient map (or identification map) if, for every subset ''U'' of ''Y'', ''U'' is open in ''Y'' If And Only If ''f'' -1(''U'') is open in ''X''. ; function, then the Quotient Topology on ''Y'' induced by ''f'' is the finest topology for which ''f'' is continuous. The space ''X'' is a quotient space or identification space. By definition, ''f'' is a quotient map. The most common example of this is to consider an Equivalence Relation on ''X'', with ''Y'' the set of Equivalence Class es and ''f'' the natural projection map. This construction is dual to the construction of the subspace topology. R ;Refinement: A cover ''K'' is a refinement of a cover ''L'' if every member of ''K'' is a subset of some member of ''L''. ; if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and ''x'' have Disjoint neighbourhoods. ; (or T3) if it is a regular T0 space. (A regular space is Hausdorff If And Only If it is T0, so the terminology is consistent.) ;Regular open: An open subset ''U'' of a space ''X'' is regular open if it equals the interior of its closure. An example of a non-regular open set is the set ''U'' = (0, 1) U (1, 2) in R with its normal topology, since 1 is in the interior of the closure of ''U'', but not in ''U''. The regular open subsets of a space form a Complete Boolean Algebra . ; in ''X'' if the closure of ''Y'' in ''X'' is compact. ;Residual: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is residual in ''X'' if the complement of ''A'' is meagre in ''X''. Also called comeagre or '''comeager'''. S ;Second category: See Meagre. ; if it has a Countable base for its topology. Every second-countable space is first-countable, separable, and Lindelöf. ; if, for every point ''x'' in ''X'', there is a neighbourhood ''U'' of ''x'' such that every loop at ''x'' in ''U'' is homotopic in ''X'' to the constant loop ''x''. Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in ''X'', whereas in the definition of locally simply connected, the homotopy must live in ''U''.) ; if it has a Countable dense subset. ; if each is Disjoint from the other's closure. ;Sequentially compact: A space is sequentially compact if every Sequence has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact. ; is a function ''f'' from ''X'' to ''Y'', such that for any points ''x'' and ''y'' in ''X'', ''d''''Y''(''f''(''x''), ''f''(''y'')) ≤ ''d''''X''(''x'', ''y''). A short map is Strictly Short if the above inequality is strict for all ''x'' and ''y'' in ''X''. ; if it is path-connected and every loop is homotopic to a Constant map. ;Smaller topology: See Coarser Topology . ;Stronger topology: See Finer Topology . Beware, some authors, especially Analyst s, use the term '''weaker topology'''. ; (or subbasis) for a topology if every non-empty proper open set in the topology is a union of Finite intersections of sets in the subbase. If ''B'' is ''any'' collection of subsets of a set ''X'', the topology on ''X'' generated by ''B'' is the smallest topology containing ''B''; this topology consists of the empty set, ''X'' and all unions of finite intersections of elements of ''B''. ;'''. ;Subcover: A cover ''K'' is a subcover (or subcovering) of a cover ''L'' if every member of ''K'' is a member of ''L''. ;Subcovering: See Subcover. ;Subspace: If ''T'' is a topology on a space ''X'', and if ''A'' is a subset of ''X'', then the Subspace Topology on ''A'' induced by ''T'' consists of all intersections of open sets in ''T'' with ''A''. This construction is dual to the construction of the quotient topology. T ; (or Kolmogorov) if for every pair of distinct points ''x'' and ''y'' in the space, either there is an open set containing ''x'' but not ''y'', or there is an open set containing ''y'' but not ''x''. ; (or Fréchet or '''accessible''') if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its Singleton s are closed. Every T1 space is T0. ;'''. ;'''. ;'''. ;'''. ;'''. ;'''. ; is the study of topologically invariant Abstract Algebra constructions on topological spaces. ; (''X'', ''T'') is a set ''X'' equipped with a collection ''T'' of subsets of ''X'' satisfying the following Axiom s: :# The empty set and ''X'' are in ''T''. :# The union of any collection of sets in ''T'' is also in ''T''. :# The intersection of any pair of sets in ''T'' is also in ''T''. :The collection ''T'' is a topology on ''X''. ;Topological sum: See Coproduct topology. ; if it is homeomorphic to a complete metric space. ;Topology: See Topological Space . ;Totally bounded: A metric space ''M'' is totally bounded if, for every ''r'' > 0, there exist a Finite cover of ''M'' by open balls of radius ''r''. A metric space is compact if and only if it is complete and totally bounded. ;Totally disconnected: A space is totally disconnected if it has no connected subset with more than one point. ; (or indiscrete topology) on a set ''X'' consists of precisely the empty set and the entire space ''X''. ; (or completely regular Hausdorff space, '''completely T3''' space, '''T3.5''' space) is a completely regular T0 space. (A completely regular space is Hausdorff If And Only If it is T0, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff. U ;Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected. ;Ultrametric: A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for all ''x'', ''y'', ''z'' in ''M'', ''d''(''x'', ''z'') ≤ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')). ;. The spaces are then said to be uniformly isomorphic and share the same Uniform Properties . ; Uniformizable /Uniformisable: A space is uniformizable if it is homeomorphic to a uniform space. ; is a set ''U'' equipped with a nonempty collection Φ of subsets of the Cartesian Product ''X'' × ''X'' satisfying the following Axiom s: |
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