On Almost T -m- continuous Multifunctions
... thatHadiJaber Mustafa and Muayad G. Mohsen [6] introduced a stronger concept than almost -continuous multifunctions, between topological spaces namely almost -continuous multifunctions (briefly, a. -c.mf.). ...
... thatHadiJaber Mustafa and Muayad G. Mohsen [6] introduced a stronger concept than almost -continuous multifunctions, between topological spaces namely almost -continuous multifunctions (briefly, a. -c.mf.). ...
Elements of Homotopy Fall 2008 Prof. Kathryn Hess Series 13 Let B
... (b) If B is path connected and E is nonempty, then p is a surjective map. Exercise 4. Let p : E−→B be a fibration, b0 ∈ B, and F := p−1 (b0 ) ⊆ E the fiber over b0 . Assume F is nonempty. Denote by i : F −→E the inclusion map. Prove the following: (a) If B is path connected, then the induced map π0 ...
... (b) If B is path connected and E is nonempty, then p is a surjective map. Exercise 4. Let p : E−→B be a fibration, b0 ∈ B, and F := p−1 (b0 ) ⊆ E the fiber over b0 . Assume F is nonempty. Denote by i : F −→E the inclusion map. Prove the following: (a) If B is path connected, then the induced map π0 ...
The θ-topology - some basic questions
... A ⊆ X. A point x ∈ X is called a θ-contact point [8] (resp. a δ-contact point [8]) of A if A ∩ cl(U) 6= ∅ (resp. A ∩ int(cl(A)) 6= ∅) for every open set U containing x. The set of all θ-contact points (resp. δ-contact points) of A ⊆ X is called the θ-closure [8] (resp. δ-closure [8]) of A and denot ...
... A ⊆ X. A point x ∈ X is called a θ-contact point [8] (resp. a δ-contact point [8]) of A if A ∩ cl(U) 6= ∅ (resp. A ∩ int(cl(A)) 6= ∅) for every open set U containing x. The set of all θ-contact points (resp. δ-contact points) of A ⊆ X is called the θ-closure [8] (resp. δ-closure [8]) of A and denot ...
File
... A subset of a topological space is open if and only if it is the neighbourhood of each of its own points. The intersection of any two neighbourhoods of a point is also its neighbourhood in a topological space. The union of any two neighbourhoods of a point is also its neighbourhood in a topological ...
... A subset of a topological space is open if and only if it is the neighbourhood of each of its own points. The intersection of any two neighbourhoods of a point is also its neighbourhood in a topological space. The union of any two neighbourhoods of a point is also its neighbourhood in a topological ...
IOSR Journal of Mathematics (IOSR-JM)
... x iI Ai. Then, x Ai for some i. Since Ai is -sg* open , there is a sg*-open set Ui such that x Ui and Ui Ai iI Ai. Hence iI Ai is a -sg* -open set. Thus s* is a topology on X. Remark 3.14: If is not regular then the above theorem is not true, that is s* is not a topology ...
... x iI Ai. Then, x Ai for some i. Since Ai is -sg* open , there is a sg*-open set Ui such that x Ui and Ui Ai iI Ai. Hence iI Ai is a -sg* -open set. Thus s* is a topology on X. Remark 3.14: If is not regular then the above theorem is not true, that is s* is not a topology ...
Pages 1-8
... subsets. Otherwise, X is called irreducible. A subset A ⊂ X is called irreducible if it is irreducible in the induced topology. Exercises 1.4. Let X be a topological space. (1) A ⊂ X is irreducible if and only if A is. (2) Let f : X → Y be a continuous map of topological spaces. If X is irreducible, ...
... subsets. Otherwise, X is called irreducible. A subset A ⊂ X is called irreducible if it is irreducible in the induced topology. Exercises 1.4. Let X be a topological space. (1) A ⊂ X is irreducible if and only if A is. (2) Let f : X → Y be a continuous map of topological spaces. If X is irreducible, ...
PDF
... Call a set X with a closure operator defined on it a closure space. Every topological space is a closure space, if we define the closure operator of the space as a function that takes any subset to its closure. The converse is also true: Proposition 1. Let X be a closure space with c the associated ...
... Call a set X with a closure operator defined on it a closure space. Every topological space is a closure space, if we define the closure operator of the space as a function that takes any subset to its closure. The converse is also true: Proposition 1. Let X be a closure space with c the associated ...
Topologies on the set of closed subsets
... one point compactification of X is embedded as a closed subset of Γ by the mapping x -»{x} with °° corresponding to 0 E Γ. When X is locally compact, the S-compact and N-compact topologies are identical, both are Hausdorff, and the monad, μ(F), of a point F E Γ is given by μ(F) = { H E * Γ | F ~ H } ...
... one point compactification of X is embedded as a closed subset of Γ by the mapping x -»{x} with °° corresponding to 0 E Γ. When X is locally compact, the S-compact and N-compact topologies are identical, both are Hausdorff, and the monad, μ(F), of a point F E Γ is given by μ(F) = { H E * Γ | F ~ H } ...
I-Sequential Topological Spaces∗
... (ii) For any topological space Y and function f : X ! Y; f is continuous if and only if it preserves I-convergence. PROOF. Suppose X is I-sequential. Any continuous function preserves I-convergence of sequences [1], so we only need to prove that if f : X ! Y preserves I-convergence, then f is contin ...
... (ii) For any topological space Y and function f : X ! Y; f is continuous if and only if it preserves I-convergence. PROOF. Suppose X is I-sequential. Any continuous function preserves I-convergence of sequences [1], so we only need to prove that if f : X ! Y preserves I-convergence, then f is contin ...
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.