CONTRA-CONTINUOUS FUNCTIONS AND STRONGLY S
... In this paper we present a new generalization of continuity called contra-continuity. We define this class of functions by the requirement that the inverse image of each open set in the codomain is closed in the domain. This notion is a stronger form of LC-continuity This definition enables us to ob ...
... In this paper we present a new generalization of continuity called contra-continuity. We define this class of functions by the requirement that the inverse image of each open set in the codomain is closed in the domain. This notion is a stronger form of LC-continuity This definition enables us to ob ...
On Colimits in Various Categories of Manifolds
... There is one subtlety which the proof above leaves out. We computed the pushout in the category of topological spaces and saw that it’s not a manifold. But how do we know that the pushout in the category of manifolds is the same as the pushout in the category of topological spaces? This is addressed ...
... There is one subtlety which the proof above leaves out. We computed the pushout in the category of topological spaces and saw that it’s not a manifold. But how do we know that the pushout in the category of manifolds is the same as the pushout in the category of topological spaces? This is addressed ...
![arXiv:math/9811003v1 [math.GN] 1 Nov 1998](http://s1.studyres.com/store/data/000335093_1-58b56f331c623a9d53167347869d2e0a-300x300.png)
Part II - Cornell Math
... (a) If there is a retraction from X to A, then i∗ : π1 (A) → π1 (X) is injective. π1 (X) is Z because π1 (S 1 × D2 ) 1 π1 (S 1 ) × π1 (D2 ) 2 π1 (S 1 ) × {1} (1 is by Thm 11.14 [L] and 2 is by Lemma 9.2 [L]). On the other hand, π1 (S 1 × S 1 ) is Z × Z. So it is impossible. (b) r : S 1 × S 1 → A ...
... (a) If there is a retraction from X to A, then i∗ : π1 (A) → π1 (X) is injective. π1 (X) is Z because π1 (S 1 × D2 ) 1 π1 (S 1 ) × π1 (D2 ) 2 π1 (S 1 ) × {1} (1 is by Thm 11.14 [L] and 2 is by Lemma 9.2 [L]). On the other hand, π1 (S 1 × S 1 ) is Z × Z. So it is impossible. (b) r : S 1 × S 1 → A ...
PDF
... δ(B) is cofinal in A, then the pair (B, γ ◦ δ) is said to be a subnet of (A, γ). Alternatively, a subnet of a net (xα )α∈A is sometimes defined to be a net (xαβ )β∈B such that for each α0 ∈ A there exists a β0 ∈ B such that αβ ≥ α0 for all β ≥ β0 . Nets are a generalisation of sequences, and in many ...
... δ(B) is cofinal in A, then the pair (B, γ ◦ δ) is said to be a subnet of (A, γ). Alternatively, a subnet of a net (xα )α∈A is sometimes defined to be a net (xαβ )β∈B such that for each α0 ∈ A there exists a β0 ∈ B such that αβ ≥ α0 for all β ≥ β0 . Nets are a generalisation of sequences, and in many ...
F-nodec spaces - RiuNet
... Definition 3.3. Let X be a topological space. X is called a ρ-nodec (resp., F H-nodec ) space if its ρ-reflection (resp., F H-reflection ) is a nodec space. In order to give a characterization of ρ-nodec spaces, we need to recall some elementary proporties which characterize Tychonoff spaces in term ...
... Definition 3.3. Let X be a topological space. X is called a ρ-nodec (resp., F H-nodec ) space if its ρ-reflection (resp., F H-reflection ) is a nodec space. In order to give a characterization of ρ-nodec spaces, we need to recall some elementary proporties which characterize Tychonoff spaces in term ...
![Forms [14 CM] and [43 W] through [43 AC] [14 CM] Kolany`s](http://s1.studyres.com/store/data/014889156_1-4ddf6cb6c42621168d150358ab1c3978-300x300.png)
Forms [14 CM] and [43 W] through [43 AC] [14 CM] Kolany`s
... In note 28 replace the first definition with the following Definition. Let (X, T ) be a topological space. 1. A set Y ⊆ X is nowhere dense if the closure of Y has empty interior. 2. A set Y ⊆ X is meager or of the first category if Y is a countable union of nowhere dense sets. 3. A set Y ⊆ X is perf ...
... In note 28 replace the first definition with the following Definition. Let (X, T ) be a topological space. 1. A set Y ⊆ X is nowhere dense if the closure of Y has empty interior. 2. A set Y ⊆ X is meager or of the first category if Y is a countable union of nowhere dense sets. 3. A set Y ⊆ X is perf ...
For printing
... directed system Δ with values in Y, and is denoted by \y \ ^ or more briefly by XJμ] 5 a directed set [yμ] converges to y (in symbols, γμ —» y) if for every neighborhood V of yj^here exists a μ' e Δ with y e F for all μ > μ'. Furthermore, we have g: Y —» Z if and only if for every directed s e t { y ...
... directed system Δ with values in Y, and is denoted by \y \ ^ or more briefly by XJμ] 5 a directed set [yμ] converges to y (in symbols, γμ —» y) if for every neighborhood V of yj^here exists a μ' e Δ with y e F for all μ > μ'. Furthermore, we have g: Y —» Z if and only if for every directed s e t { y ...
rg\alpha-closed sets and rg\alpha
... τ -int (A), sint (A), scl (A), w-int (A), w-cl (A), gpr-int (A), gpr-cl (A), αint (A), α-cl (A) and AC or X − A respectively. (X, τ ) will be replaced by X if there is no chance of confusion. Let us recall the following definitions as pre requesties. Definition 1.1. A subset A of a space X is called ...
... τ -int (A), sint (A), scl (A), w-int (A), w-cl (A), gpr-int (A), gpr-cl (A), αint (A), α-cl (A) and AC or X − A respectively. (X, τ ) will be replaced by X if there is no chance of confusion. Let us recall the following definitions as pre requesties. Definition 1.1. A subset A of a space X is called ...