Metric and Topological Spaces T. W. K¨orner October 16, 2014
... Some definitions and results transfer essentially unchanged from classical analysis on R to metric spaces. Recall the classical definition of continuity. Definition 4.1. [Old definition.] A function f : R → R is called continuous if, given t ∈ R and ǫ > 0, we can find a δ(t, ǫ) > 0 such that |f (t) ...
... Some definitions and results transfer essentially unchanged from classical analysis on R to metric spaces. Recall the classical definition of continuity. Definition 4.1. [Old definition.] A function f : R → R is called continuous if, given t ∈ R and ǫ > 0, we can find a δ(t, ǫ) > 0 such that |f (t) ...
Introductory notes in topology
... with respect to the topology, such that the empty set ∅ and X itself are open sets, the intersection of finitely many open sets is an open set, and the union of any family of open sets is an open set. For any set X, the indiscrete topology has ∅, X as the only open sets, and the discrete topology is ...
... with respect to the topology, such that the empty set ∅ and X itself are open sets, the intersection of finitely many open sets is an open set, and the union of any family of open sets is an open set. For any set X, the indiscrete topology has ∅, X as the only open sets, and the discrete topology is ...
Semi-quotient mappings and spaces
... the pre-image of B. By Cl.A/ and Int.A/ we denote the closure and interior of a set A in a space X . Our other topological notation and terminology are standard as in [10]. If .G; / is a group, then e or eG denotes its identity element, and for a given x 2 G, `x W G ! G, y 7! x ı y, and rx W G ! G, ...
... the pre-image of B. By Cl.A/ and Int.A/ we denote the closure and interior of a set A in a space X . Our other topological notation and terminology are standard as in [10]. If .G; / is a group, then e or eG denotes its identity element, and for a given x 2 G, `x W G ! G, y 7! x ı y, and rx W G ! G, ...
Maximal Tychonoff Spaces and Normal Isolator Covers
... Vx ⊂ Ux . Therefore the cover W = {Vx : x ∈ X} is the required open cover of (X, τ ), which is a refinement of U. (ii) ⇒ (i): Let the condition holds and if possible let τ1 be a Tychonoff (or uniformizable) topology that contains no isolated points satisfying τ ⊂ τ1 . It is sufficient to prove that ...
... Vx ⊂ Ux . Therefore the cover W = {Vx : x ∈ X} is the required open cover of (X, τ ), which is a refinement of U. (ii) ⇒ (i): Let the condition holds and if possible let τ1 be a Tychonoff (or uniformizable) topology that contains no isolated points satisfying τ ⊂ τ1 . It is sufficient to prove that ...
ON θ-PRECONTINUOUS FUNCTIONS
... if the preimage f −1 (V ) of each open set V of Y is preopen in X. Precontinuity was called near continuity by Pták [26] and also called almost continuity by Frolík [9] and Husain [10]. In 1985, Janković [12] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [23] int ...
... if the preimage f −1 (V ) of each open set V of Y is preopen in X. Precontinuity was called near continuity by Pták [26] and also called almost continuity by Frolík [9] and Husain [10]. In 1985, Janković [12] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [23] int ...
Open subgroups and Pontryagin duality
... P r o o f The discrete group G/A admits sufficiently many continuous'characters, which proves (a). For (b), see [4, Lemma 3.3]. Statement (c) follows from (1.2). Now, (b) says that # " G ---,A^ is surjective. To prove that #~ is open, take an arbitrary compact subset X of G. In view of (1.1), we onl ...
... P r o o f The discrete group G/A admits sufficiently many continuous'characters, which proves (a). For (b), see [4, Lemma 3.3]. Statement (c) follows from (1.2). Now, (b) says that # " G ---,A^ is surjective. To prove that #~ is open, take an arbitrary compact subset X of G. In view of (1.1), we onl ...
NON-MEAGER P-FILTERS ARE COUNTABLE DENSE
... filter, there exists x ∈ F such that ω − x is infinite. Thus, {y : x ⊂ y ⊂ ω} is a copy of the Cantor set contained in F. Further, ω F always contains a copy of the Cantor set. So it is always true that X contains a copy C of the Cantor set. Assume that F is meager, let us arrive to a contradiction. ...
... filter, there exists x ∈ F such that ω − x is infinite. Thus, {y : x ⊂ y ⊂ ω} is a copy of the Cantor set contained in F. Further, ω F always contains a copy of the Cantor set. So it is always true that X contains a copy C of the Cantor set. Assume that F is meager, let us arrive to a contradiction. ...
