An Internal Solution to the Problem of Linearization of a Convexity Space

1976 ◽  
Vol 19 (4) ◽  
pp. 487-494 ◽  
Author(s):  
D. A. Szafron ◽  
J. H. Weston
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Linear Structure ◽  
Linear Topological Space ◽  
Necessary And Sufficient Conditions ◽  
Convexity Space ◽  
Internal Solution ◽  
Convexity Structure ◽  
Necessary And Sufficient

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

scholarly journals On a topology between Tα and Tγα

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3209-3221
Author(s):  
Dimitrije Andrijevic
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closure Operators ◽  
New Class ◽  
Necessary And Sufficient

Using the topology T in a topological space (X,T), a new class of generalized open sets called ?-preopen sets, is introduced and studied. This class generates a new topology Tg which is larger than T? and smaller than T??. By means of the corresponding interior and closure operators, among other results, necessary and sufficient conditions are given for Tg to coincide with T? , T? or T??.

scholarly journals A characterization of point semiuniformities

1996 ◽  
Vol 19 (2) ◽  
pp. 311-316
Author(s):  
Jennifer P. Montgomery
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Groups ◽  
Necessary And Sufficient

The concept of a uniformity was developed by A. Well and there have been several generalizations. This paper defines a point semiuniformity and gives necessary and sufficient conditions for a topological space to be point semiuniformizable. In addition, just as uniformities are associated with topological groups, a point semiuniformity is naturally associated with a semicontinuous group. This paper shows that a point semiuniformity associated with a semicontinuous group is a uniformity if and only if the group is a topological group.

The volume of certain convex sets

1982 ◽  
Vol 91 (1) ◽  
pp. 91-97 ◽  
Author(s):  
P. McMullen
Keyword(s):  
Convex Body ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Subspaces ◽  
Necessary And Sufficient

AbstractLet Ll …, Lr be independent linear subspaces of Ed, with di = dim Li ≥ 1 (i = 1, …, r), and Ed = L1 + … + Lr. For each i, let K¯;i be a convex body in Li with 0 ∈ K¯i, Ki = K¯i + ti(ti ∈ Ed) any translate of K¯i, and define K = con v (K1 ∪ … ∪ Kr), and similarl K¯. Then vol K ≥ vol K¯. Necessary and sufficient conditions for equality are also obtained.

scholarly journals Near-ring-semigroups of continuous selfmaps

1988 ◽  
Vol 37 (2) ◽  
pp. 277-291 ◽  
Author(s):  
K.D. Magill
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analogous Problem ◽  
Multiplicative Semigroup ◽  
Necessary And Sufficient

We find necessary and sufficient conditions on a topological space X so that S(X), the semigroup of all continuous selfmaps of X, is isomorphic to the multiplicative semigroup of a near-ring. The analogous problem is also considered for the semigroup of all continuous selfmaps which fix some point of X.

scholarly journals The largest proper congruence onS(X)

1984 ◽  
Vol 7 (4) ◽  
pp. 663-666 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

S(X)denotes the semigroup of all continuous selfmaps of the topological spaceX. In this paper, we find, for many spacesX, necessary and sufficient conditions for a certain type of congruence to be the largest proper congruence onS(X).

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

scholarly journals Continuous Representability of Interval Orders: The Topological Compatibility Setting

2015 ◽  
Vol 23 (03) ◽  
pp. 345-365 ◽  
Author(s):  
G. Bosi ◽  
A. Estevan ◽  
J. Gutiérrez García ◽  
E. Induráin
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Interval Order ◽  
Topological Spaces ◽  
Interval Orders ◽  
Finite Support ◽  
Order Representation ◽  
Necessary And Sufficient ◽  
The Given

In this paper, we go further on the problem of the continuous numerical representability of interval orders defined on topological spaces. A new condition of compatibility between the given topology and the indifference associated to the main trace of an interval order is introduced. Provided that this condition is fulfilled, a semiorder has a continuous interval order representation through a pair of continuous real-valued functions. Other necessary and sufficient conditions for the continuous representability of interval orders are also discussed, and, in particular, a characterization is achieved for the particular case of interval orders defined on a topological space of finite support.

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