Limit Points and Closure
... element of A. This is clearly false and so we have a contradiction to our supposition. Therefore every limit point of A must belong to A. Conversely, assume that A contains all of its limit points. For each z X \A, our assumption implies that there exists an open set U z 3 z such that U z A = Ø; ...
... element of A. This is clearly false and so we have a contradiction to our supposition. Therefore every limit point of A must belong to A. Conversely, assume that A contains all of its limit points. For each z X \A, our assumption implies that there exists an open set U z 3 z such that U z A = Ø; ...
COMBINATORIAL HOMOTOPY. I 1. Introduction. This is the first of a
... Let K be the universal covering complex of a CW-complex K. Since TI(K) = 1, irn(K) ^Tn(K) if n> 1 and since L is a Jm-complex if 7TV(L) = 0 for r = 1, • • • , m — 1 it follows from Theorem 10 that K is a J m -complex if 7r n (i£)=0 for n = 2, • • • , m — 1 . In particular if is a J^-complex if its u ...
... Let K be the universal covering complex of a CW-complex K. Since TI(K) = 1, irn(K) ^Tn(K) if n> 1 and since L is a Jm-complex if 7TV(L) = 0 for r = 1, • • • , m — 1 it follows from Theorem 10 that K is a J m -complex if 7r n (i£)=0 for n = 2, • • • , m — 1 . In particular if is a J^-complex if its u ...
K - CIS @ UPenn
... It should be noted that for a 0-simplex consisting of a single point {a0}, ∂{a0} = ∅, and Int {a0} = {a0}. Of course, a 0-simplex is a single point, a 1-simplex is the line segment (a0, a1), a 2-simplex is a triangle (a0, a1, a2) (with its interior), and a 3-simplex is a tetrahedron (a0, a1, a2, a3) ...
... It should be noted that for a 0-simplex consisting of a single point {a0}, ∂{a0} = ∅, and Int {a0} = {a0}. Of course, a 0-simplex is a single point, a 1-simplex is the line segment (a0, a1), a 2-simplex is a triangle (a0, a1, a2) (with its interior), and a 3-simplex is a tetrahedron (a0, a1, a2, a3) ...
AN APPLICATION OF MACKEY`S SELECTION LEMMA 1
... Therefore the map (r, d) is open Corollary 1. Let G be a locally compact groupoid having open range map. Let F be a subset of G(0) meeting each orbit exactly once. If the map dF : GF → G(0) , dF (x) = d (x), is open, then the orbit space G(0) /G is proper. Proof. The fact that G(0) /G is a proper sp ...
... Therefore the map (r, d) is open Corollary 1. Let G be a locally compact groupoid having open range map. Let F be a subset of G(0) meeting each orbit exactly once. If the map dF : GF → G(0) , dF (x) = d (x), is open, then the orbit space G(0) /G is proper. Proof. The fact that G(0) /G is a proper sp ...
On Totally sg-Continuity, Strongly sg
... Theorem 4.6 Assume that arbitrary union of sg-open sets is sg-open. If X is a topological space and for each pair of distinct points x1 and x2 in X there exists a map f into a Urysohn topological space Y such that f(x1) ≠ f(x2) and f is csgcontinuous at x1 and x2, then X is sg-T2. Proof. Let x1 and ...
... Theorem 4.6 Assume that arbitrary union of sg-open sets is sg-open. If X is a topological space and for each pair of distinct points x1 and x2 in X there exists a map f into a Urysohn topological space Y such that f(x1) ≠ f(x2) and f is csgcontinuous at x1 and x2, then X is sg-T2. Proof. Let x1 and ...
