Topological duality theorems. II. Non-closed sets

1963 ◽  
pp. 103-233
Author(s):  
P. S. Aleksandrov
Keyword(s):  
Duality Theorems ◽  
Topological Duality ◽  
Closed Sets

The Dissection of Closed Sets of Arbitrary Dimension and the Generalized Brouwer-Alexandroff Theorem

1935 ◽  
Vol 31 (4) ◽  
pp. 525-535 ◽  
Author(s):  
B. Kaufmann
Keyword(s):  
Local Structure ◽  
Arbitrary Dimension ◽  
Geometrical Theory ◽  
Duality Theorems ◽  
Successful Development ◽  
Closed Sets ◽  
Infinite Cycles

The successful development in recent years of the topology of general closed sets is largely due to the application of combinatory methods, which have led to an elaborate theory of approximation of closed sets by infinite cycles, to the generalization of duality theorems for closed sets, and to a geometrical theory of dimensions. Corresponding to the combinatory invariants involved the main results of these theories concern for the most part properties of the set as a whole, which cannot possibly express fully its internal and in particular its local structure. Still less is our knowledge of the eventual relations between the local structure of a set and its properties in the large.

GEOMETRICAL AND TOPOLOGICAL DUALITY IN CROWD DYNAMICS

2010 ◽  
Vol 03 (04) ◽  
pp. 493-507 ◽  
Author(s):  
V. G. IVANCEVIC ◽  
D. J. REID
Keyword(s):  
Algebraic Topology ◽  
Duality Theorem ◽  
Geometrical Model ◽  
Related Question ◽  
Crowd Behavior ◽  
Duality Theorems ◽  
Topological Duality ◽  
Crowd Dynamics ◽  
Behavior Dynamics ◽  
Crowd Behavior Dynamics

The purpose of this paper is to establish strong theoretical basis for solving practical problems in modeling the behavior of crowds. Based on the previously developed (entropic) geometrical model of crowd behavior dynamics, in this paper we formulate two duality theorems related to the crowd manifold. Firstly, we formulate the geometrical crowd-duality theorem and prove it using Lie-functorial and Riemannian proofs. Secondly, we formulate the topological crowd-duality theorem and prove it using cohomological and homological proofs. After that we discuss the related question of the connection between Lagrangian and Hamiltonian crowd-duality, and finally establish the globally dual structure of crowd dynamics. All used terms from algebraic topology are defined in Appendix A.

Generalized ʎ-Closed Sets and (ʎ-ʏ)*-Continuous Function

2017 ◽  
Vol 4 (ICBS Conference) ◽  
pp. 1-17 ◽  
Author(s):  
Alias Khalaf ◽  
Sarhad Nami
Keyword(s):  
Continuous Function ◽  
Closed Sets

ON ir-CLOSED SETS IN TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (5) ◽  
pp. 2573-2582
Author(s):  
A. M. Anto ◽  
G. S. Rekha ◽  
M. Mallayya
Keyword(s):  
Topological Spaces ◽  
Closed Sets

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

ON CONTRA SOFT ARS CLOSED SETS IN SOFT TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (3) ◽  
pp. 921-926
Author(s):  
P. Anbarasi Rodrigo ◽  
K. Rajendra Suba
Keyword(s):  
Topological Spaces ◽  
Closed Sets
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