Metric and Banach spaces
... Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) → (Y, dY ). If (xn )n∈N is a Cauchy sequence of X, then f (xn )n∈N is a Cauchy sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about conver ...
... Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) → (Y, dY ). If (xn )n∈N is a Cauchy sequence of X, then f (xn )n∈N is a Cauchy sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about conver ...
on spaces whose nowhere dense subsets are scati`ered 1
... THEOREM 2.2. If (X, v) is a T dense-in-itself space, then X is N-scattered N(r) S(r), where N(T) is the ideal of nowhere dense subsets of X, and S(r) is the family of scattered subsets of X. PROOF. It is known that in a T-space, S(’) is the smallest local ideal containing the ideal I of finite sets ...
... THEOREM 2.2. If (X, v) is a T dense-in-itself space, then X is N-scattered N(r) S(r), where N(T) is the ideal of nowhere dense subsets of X, and S(r) is the family of scattered subsets of X. PROOF. It is known that in a T-space, S(’) is the smallest local ideal containing the ideal I of finite sets ...
For printing - Mathematical Sciences Publishers
... countable spaces is pseudocompact. Since there is no reason, in general, to expect such products to be ^-pseudocompact, one cannot hope for a result analogous to Theorem 2.6 for pseudocompact powers. This can be seen in another way. In [11], Glicksberg shows that a product space ΐlaeiXa is pseudocom ...
... countable spaces is pseudocompact. Since there is no reason, in general, to expect such products to be ^-pseudocompact, one cannot hope for a result analogous to Theorem 2.6 for pseudocompact powers. This can be seen in another way. In [11], Glicksberg shows that a product space ΐlaeiXa is pseudocom ...
S -paracompactness in ideal topological spaces
... {Cl (Hλ ) : λ ∈ Λ} = X − {Cl (Wλ ) : λ ∈ Λ} ⊂ X − {Wλ : λ ∈ Λ} ∈ I. Thus, H = {Cl (Hλ ) : λ ∈ Λ} is an I-cover of X. Corollary 2.3. If (X, τ, I) is an e.d. regular I-S-paracompact space, then (X, τ, I) is I-paracompact. Proof. Let U be an open cover of X, using the I-S-paracompactness of (X, τ, I), ...
... {Cl (Hλ ) : λ ∈ Λ} = X − {Cl (Wλ ) : λ ∈ Λ} ⊂ X − {Wλ : λ ∈ Λ} ∈ I. Thus, H = {Cl (Hλ ) : λ ∈ Λ} is an I-cover of X. Corollary 2.3. If (X, τ, I) is an e.d. regular I-S-paracompact space, then (X, τ, I) is I-paracompact. Proof. Let U be an open cover of X, using the I-S-paracompactness of (X, τ, I), ...
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.