Closed graph theorems and Baire spaces
... The answer is “no”. In [16] an example is given of a nearly continuous function with closed graph acting from a metric space into a complete metric space which is not continuous. Incidentally, this function provides an example of a very nearly continuous function that is not continuous, (see the pro ...
... The answer is “no”. In [16] an example is given of a nearly continuous function with closed graph acting from a metric space into a complete metric space which is not continuous. Incidentally, this function provides an example of a very nearly continuous function that is not continuous, (see the pro ...
A Note on Free Topological Groupoids
... [lG] and MORRIS[12, 13, 141. It should be noted that our proof depends heavily on the work of BROWN and HARDY[ 2 ] . They proved that (They did not show that for any k<,,-topologicalgraph r, P(r)is HAUSDORFF. i: l’+P(l‘) is an embedding). Finally we record that our proof, even when specialiized to t ...
... [lG] and MORRIS[12, 13, 141. It should be noted that our proof depends heavily on the work of BROWN and HARDY[ 2 ] . They proved that (They did not show that for any k<,,-topologicalgraph r, P(r)is HAUSDORFF. i: l’+P(l‘) is an embedding). Finally we record that our proof, even when specialiized to t ...
ALGEBRAIC TOPOLOGY Contents 1. Preliminaries 1 2. The
... notion explained earlier of ‘wrapping’ R around S 1 as a helix. Choose our x̃0 to be 0 ∈ R, and let x0 = (1, 0). We thus have that p−1 (1, 0) = Z. Since R is simply connected, the lifting correspondence Φ : π1 (S 1 , x0 ) → Z is bijective according to our above proposition. We now need to prove that ...
... notion explained earlier of ‘wrapping’ R around S 1 as a helix. Choose our x̃0 to be 0 ∈ R, and let x0 = (1, 0). We thus have that p−1 (1, 0) = Z. Since R is simply connected, the lifting correspondence Φ : π1 (S 1 , x0 ) → Z is bijective according to our above proposition. We now need to prove that ...
CONNECTIVE SPACES 1. Connective Spaces 1.1. Introduction. As
... 1.1. Introduction. As a topological concept connectedness is of somewhat different character than most other important properties, such as the covering properties, studied in the category TOP. Its aim is to topologically explain the intuitive notion of continuity of a point set. Roughly speaking, a ...
... 1.1. Introduction. As a topological concept connectedness is of somewhat different character than most other important properties, such as the covering properties, studied in the category TOP. Its aim is to topologically explain the intuitive notion of continuity of a point set. Roughly speaking, a ...
Some Properties of Contra-b-Continuous and Almost Contra
... Proof. Let V be any regular closed set on Y . Then since f is almost contra-bcontinuous and almost continuous, then by Theorem 3.9 f is R-map. Hence f −1 (V ) is regular closed in X . Let {Vα : α ∈ I} be any regular closed cover of Y . Then { f −1 (Vα ) : α ∈ I} is a regular closed cover of X and si ...
... Proof. Let V be any regular closed set on Y . Then since f is almost contra-bcontinuous and almost continuous, then by Theorem 3.9 f is R-map. Hence f −1 (V ) is regular closed in X . Let {Vα : α ∈ I} be any regular closed cover of Y . Then { f −1 (Vα ) : α ∈ I} is a regular closed cover of X and si ...
The Cantor Discontinuum
... is a Cantor discontinuum and if D ⊂ R, then card D = card R = 2ℵ0 . 3. Cantor’s Middle Third Set K ⊂ [0, 1] ⊂ R has Lebesgue measure zero. 4. There exist Cantor discontinuums in R of positive Lebesgue measure. In fact, for every 0 ≤ < 1, there exist Cantor discontinuums D ⊂ [0, 1] of Lebesgue meas ...
... is a Cantor discontinuum and if D ⊂ R, then card D = card R = 2ℵ0 . 3. Cantor’s Middle Third Set K ⊂ [0, 1] ⊂ R has Lebesgue measure zero. 4. There exist Cantor discontinuums in R of positive Lebesgue measure. In fact, for every 0 ≤ < 1, there exist Cantor discontinuums D ⊂ [0, 1] of Lebesgue meas ...
Multifunctions and graphs - Mathematical Sciences Publishers
... (c ) If Ω is a filterbase on X with Q —> θ% in X then ad* Ω c Φ(x). (d) If Ω is a filterbase on X with Ω—>θx in X then y eΦ(x) whenever Ω* —> θy and ώ* is finer than Φ(Ω). ( e ) // {xn} and {yn} are nets on X and Y, respectively, with xn-*θ% in X, yn-*θV in Y and yneΦ{xn) for each n, then yeΦ{x). (f ...
... (c ) If Ω is a filterbase on X with Q —> θ% in X then ad* Ω c Φ(x). (d) If Ω is a filterbase on X with Ω—>θx in X then y eΦ(x) whenever Ω* —> θy and ώ* is finer than Φ(Ω). ( e ) // {xn} and {yn} are nets on X and Y, respectively, with xn-*θ% in X, yn-*θV in Y and yneΦ{xn) for each n, then yeΦ{x). (f ...
Extension of continuous functions in digital spaces with the
... of all translates A + c with c1 , c2 ∈ 2Z. This family is a subbasis for the topology. The family of all intersections of these sets is a basis, and the ...
... of all translates A + c with c1 , c2 ∈ 2Z. This family is a subbasis for the topology. The family of all intersections of these sets is a basis, and the ...
Examples of random groups - Irma
... Step II – Probability and harmonic analysis. It remains to show the existence of a labelling required at each inductive step and of a subset I ⊆ N such that ΘI admits the iteration. We obtain the former in Section 5. We endow the space of all possible labellings of Θn (and hence, those of Θ) with th ...
... Step II – Probability and harmonic analysis. It remains to show the existence of a labelling required at each inductive step and of a subset I ⊆ N such that ΘI admits the iteration. We obtain the former in Section 5. We endow the space of all possible labellings of Θn (and hence, those of Θ) with th ...
Branched coverings
... sufficient for most topological applications. The most common use is in attempts to classify manifolds. By a well-known classical theorem of Alexander [1], each closed orientable PL n-manifold can be obtained as a branched covering over S". In dimension 3, which is of special interest, the branched ...
... sufficient for most topological applications. The most common use is in attempts to classify manifolds. By a well-known classical theorem of Alexander [1], each closed orientable PL n-manifold can be obtained as a branched covering over S". In dimension 3, which is of special interest, the branched ...
Analogues of Cayley graphs for topological groups
... groups to the rough ends. As a byproduct we deduce a result due to Dunwoody and Roller [11] (also proved by Niblo [43] and Scott and Swarup [49]). In Section 4 the growth of the graph X is related to the growth of the topological group G. The outcome is a version of Gromov’s theorem on groups of pol ...
... groups to the rough ends. As a byproduct we deduce a result due to Dunwoody and Roller [11] (also proved by Niblo [43] and Scott and Swarup [49]). In Section 4 the growth of the graph X is related to the growth of the topological group G. The outcome is a version of Gromov’s theorem on groups of pol ...
A New Type of Weak Continuity 1 Introduction
... Lemma 4.2 The function f : (X, τ ) → (Y, σ) has a ultra sgα-closed graph if and only if for every (x, y) ∈ (X × Y )\G(f ) there exist U ∈ sgαO(X, x), V ∈ sgαO(Y, y) and f (U ) ∩ sgα-Cl(V ) = ∅. Proof : It is an immediate consequence of Definition 4.1. Theorem 4.3 Let f : (X, τ ) → (Y, σ) be a weakly ...
... Lemma 4.2 The function f : (X, τ ) → (Y, σ) has a ultra sgα-closed graph if and only if for every (x, y) ∈ (X × Y )\G(f ) there exist U ∈ sgαO(X, x), V ∈ sgαO(Y, y) and f (U ) ∩ sgα-Cl(V ) = ∅. Proof : It is an immediate consequence of Definition 4.1. Theorem 4.3 Let f : (X, τ ) → (Y, σ) be a weakly ...
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... Corollary 3.4 (See [10]). Let X, Y be topological spaces and let X be Čechcomplete. Let F : Y → X be a compact-valued mapping with a closed graph. Then the set of points of upper semicontinuity of F is a Gδ subset of Y . Corollary 3.5. Let X, Y be topological spaces and let X be locally compact. Let ...
... Corollary 3.4 (See [10]). Let X, Y be topological spaces and let X be Čechcomplete. Let F : Y → X be a compact-valued mapping with a closed graph. Then the set of points of upper semicontinuity of F is a Gδ subset of Y . Corollary 3.5. Let X, Y be topological spaces and let X be locally compact. Let ...
Reconstructing a Simple Polygon from Its Angles
... every vertex may seem to be far too much information and the reconstruction problem may thus seem easily solvable by some greedy algorithm. Before we actually present the triangle witness algorithm that solves the reconstruction problem, we show that some natural greedy algorithms do not work in gen ...
... every vertex may seem to be far too much information and the reconstruction problem may thus seem easily solvable by some greedy algorithm. Before we actually present the triangle witness algorithm that solves the reconstruction problem, we show that some natural greedy algorithms do not work in gen ...
Ordered Pairs - Hempfield Curriculum
... 2. If what Marcel thinks about his quadrilateral is true, what type of quadrilateral does he have? 3. Richelle drew hexagon KLMNOP at the right. She thinks the hexagon has six congruent angles. How can she show that the angles are congruent without using a protractor to measure them? ...
... 2. If what Marcel thinks about his quadrilateral is true, what type of quadrilateral does he have? 3. Richelle drew hexagon KLMNOP at the right. She thinks the hexagon has six congruent angles. How can she show that the angles are congruent without using a protractor to measure them? ...
Italo Jose Dejter
Italo Jose Dejter (Bahia Blanca, 1939) is an Argentine-born American mathematician of Bessarabian Jewish descent. He is a professor at the University of Puerto Rico (UPRRP) since 1984 and has conducted research in mathematics, particularly in areas that include algebraic topology, differential topology, graph theory, coding theory and design theory.He has an Erdos number of 2 since 1993.Dejter completed the Licentiate degree in Mathematics from University of Buenos Aires in 1967, and the Ph.D. degree in Mathematics from Rutgers University in 1975 under the supervision of Ted Petrie. He was a professor atFederal University of Santa Catarina, Brazil, from 1977 to 1984. Dejter has been a visiting scholar at a number of research institutions, including University of São Paulo,Instituto Nacional de Matemática Pura e Aplicada, Federal University of Rio Grande do Sul,University of Cambridge,National Autonomous University of Mexico, Simon Fraser University, University of Victoria, New York University, University of Illinois at Urbana–Champaign, McMaster University,DIMACS, Autonomous University of Barcelona, Technical University of Denmark, Auburn University, Polytechnic University of Catalonia, Technical University of Madrid, Charles University, Ottawa University, Simón Bolívar University, etc. The sections below describe the relevance of Dejter's work in the research areas mentioned in the first paragraph above, or in the box to the right.