strongly connected spaces - National University of Singapore
... between f(a) and f(b), then there exists an element c ∈ [a, b] such that f(c) = r. This theorem is used in a number of places, for instance when constructing inverse functions such as sin-1(x). The property of the space [a, b] on which the Intermediate Value Theorem depends is the connectedness, and ...
... between f(a) and f(b), then there exists an element c ∈ [a, b] such that f(c) = r. This theorem is used in a number of places, for instance when constructing inverse functions such as sin-1(x). The property of the space [a, b] on which the Intermediate Value Theorem depends is the connectedness, and ...
On productively Lindelöf spaces
... Aurichi and the first named author proved that a metrizable space is indestructibly productively Lindelöf if and only if it is σ -compact [9]. It is easily seen that if a space Y is Hurewicz in a countably closed extension, then it is Hurewicz. The following theorem answers a question of Aurichi and ...
... Aurichi and the first named author proved that a metrizable space is indestructibly productively Lindelöf if and only if it is σ -compact [9]. It is easily seen that if a space Y is Hurewicz in a countably closed extension, then it is Hurewicz. The following theorem answers a question of Aurichi and ...
sA -sets and decomposition of sA
... 1. A is an sCI -set and a semi∗ -I-open set in X. 2. A = L ∩ cl(int∗ (A)) for a semi-I-open set L. Proof. (1) ⇒ (2): Suppose that A is an sCI -set and a semi∗ -I-open set in X. Since A is sCI -set, then we have A = L ∩ M, where L ∈ SIO(X) and M is a pre-I-closed set in X. We have A ⊆ M, so cl(int∗ ( ...
... 1. A is an sCI -set and a semi∗ -I-open set in X. 2. A = L ∩ cl(int∗ (A)) for a semi-I-open set L. Proof. (1) ⇒ (2): Suppose that A is an sCI -set and a semi∗ -I-open set in X. Since A is sCI -set, then we have A = L ∩ M, where L ∈ SIO(X) and M is a pre-I-closed set in X. We have A ⊆ M, so cl(int∗ ( ...
one-point compactification on convergence spaces
... PROOF. Let x E X* and h.t .s lc ;t net in X* such that every universal subnet of s converges tox inX* Supposex isinX. Let u bcauniversalsubnet ofs. Sinceu-x, uis frequently in X and ulX x. By Lemma 1.3, u is eventually in X. According to Theorem 1.4, s is frequently in X. Let v bc a univcrsd subnct ...
... PROOF. Let x E X* and h.t .s lc ;t net in X* such that every universal subnet of s converges tox inX* Supposex isinX. Let u bcauniversalsubnet ofs. Sinceu-x, uis frequently in X and ulX x. By Lemma 1.3, u is eventually in X. According to Theorem 1.4, s is frequently in X. Let v bc a univcrsd subnct ...