Topology HW11,5
... The topological space X is said to be connected if X is not the union of two disjoint nonempty open subsets. Note that X is connected if only if Ø and X are the only clopen subsets of X. A subset A of X is called connected if A is a connected space with its induced topology. Clearly, a singleton sub ...
... The topological space X is said to be connected if X is not the union of two disjoint nonempty open subsets. Note that X is connected if only if Ø and X are the only clopen subsets of X. A subset A of X is called connected if A is a connected space with its induced topology. Clearly, a singleton sub ...
On $\ alpha $-continuous functions
... The notion of a ^-continuous function between topological spaces was introduced by S. Fomin in his study of extensions of Hausdorff spaces [6]. Since then this concept has been frequently used in investigations of nonregular Hausdorff spaces. A function / : X —• Y is called ^-continuous if for every ...
... The notion of a ^-continuous function between topological spaces was introduced by S. Fomin in his study of extensions of Hausdorff spaces [6]. Since then this concept has been frequently used in investigations of nonregular Hausdorff spaces. A function / : X —• Y is called ^-continuous if for every ...
... λ -open subset U x X containing x such that U x A is countable. The complement of an ω - λ -open subset is said to be ω - λ -closed. Proposition 2.2: Every λ -open set is ω - λ -open. Converse not true. Corollary 2.3: Every open set is ω - λ -open, but not conversely. Proof: Follows from the fac ...
C. Carpintero, N. Rajesh, E. Rosas and S. Saranyasri
... Proposition 2.23. Let f : (X, τ ) → (Y, σ) be somewhat ω-open and A be any open subset of X. Then f|A : (A, τ|A ) → (Y, σ) is somewhat ω-open. Proof. Let U ∈ τ|A such that U 6= ∅. Since U is open in A and A is open in X, U is open in X and since by hypothesis f : (X, τ ) → (Y, σ) is somewhat ω-open ...
... Proposition 2.23. Let f : (X, τ ) → (Y, σ) be somewhat ω-open and A be any open subset of X. Then f|A : (A, τ|A ) → (Y, σ) is somewhat ω-open. Proof. Let U ∈ τ|A such that U 6= ∅. Since U is open in A and A is open in X, U is open in X and since by hypothesis f : (X, τ ) → (Y, σ) is somewhat ω-open ...
Introduction to Topology
... Lemma 13.2. Let (X , T ) be a topological space. Suppose that C is a collection of open sets of X such that for each open subset U ⊂ X and each x ∈ U, there is an element C ∈ C such that x ∈ C ⊂ U. Then C is a basis for the topology T on X . Proof (continued). Let T 0 be the topology on X generated ...
... Lemma 13.2. Let (X , T ) be a topological space. Suppose that C is a collection of open sets of X such that for each open subset U ⊂ X and each x ∈ U, there is an element C ∈ C such that x ∈ C ⊂ U. Then C is a basis for the topology T on X . Proof (continued). Let T 0 be the topology on X generated ...
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.