How to find a Khalimsky-continuous approximation of a real-valued function Erik Melin
... even point is closed and that an odd point is open. In terms of smallest neighborhoods, we have N (m) = {m} if m is odd and N (n) = {n ± 1, n} if n is even. Let a and b, a 6 b, be integers. A Khalimsky interval is an interval [a, b] ∩ Z of integers with the topology induced from the Khalimsky line. ...
... even point is closed and that an odd point is open. In terms of smallest neighborhoods, we have N (m) = {m} if m is odd and N (n) = {n ± 1, n} if n is even. Let a and b, a 6 b, be integers. A Khalimsky interval is an interval [a, b] ∩ Z of integers with the topology induced from the Khalimsky line. ...
fixed points and admissible sets
... Theorem 2. Let X be a bounden general convex topological space with general convex structure G(x, y, λ) satisfying conditions (1) and (2). Let F = {T | T : X → X} be a finite commuting family of nonexpansive mappings of X into X than F has a common fixed point. Proof. Let Φ be a family of all nonempty ...
... Theorem 2. Let X be a bounden general convex topological space with general convex structure G(x, y, λ) satisfying conditions (1) and (2). Let F = {T | T : X → X} be a finite commuting family of nonexpansive mappings of X into X than F has a common fixed point. Proof. Let Φ be a family of all nonempty ...
Sober Spaces, Well-Filtration and Compactness Principles
... As usual, we denote the Scott topology (consisting of all Scott-open sets) of a poset P by σP , and the resulting space by ΣP . The Lawson dual P̂ = δP is the set of all Scott-open filters, ordered by inclusion. It plays a central role in the duality theory for continuous posets and semilattices (se ...
... As usual, we denote the Scott topology (consisting of all Scott-open sets) of a poset P by σP , and the resulting space by ΣP . The Lawson dual P̂ = δP is the set of all Scott-open filters, ordered by inclusion. It plays a central role in the duality theory for continuous posets and semilattices (se ...
Professor Smith Math 295 Lecture Notes
... is a subcover of S. We make the following formal definition: Definition: If {Uλ }λ∈Λ is a collection of open sets, a subcollection is any collection of fewer sets from the collection. We can write it {Uλ }λ∈Λ0 where Λ0 is some subset of the indexing set Λ. If the collection {Uλ }λ∈Λ is an open cover ...
... is a subcover of S. We make the following formal definition: Definition: If {Uλ }λ∈Λ is a collection of open sets, a subcollection is any collection of fewer sets from the collection. We can write it {Uλ }λ∈Λ0 where Λ0 is some subset of the indexing set Λ. If the collection {Uλ }λ∈Λ is an open cover ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.