Closure, Interior and Compactness in Ordinary Smooth Topological
... (OST3 ) τ ( α∈Γ Aα ) ≥ α∈Γ τ (Aα ), ∀{Aα } ⊂ 2X . The pair (X, τ ) is called an ordinary smooth topological space (in short, osts). We will denote the set of all ost’s on X as OST(X). Remark 2.2. Ying [8] called the mapping τ : 2X → I [resp. τ : I X → 2 and τ : I X → I] satisfying the axioms in Defi ...
... (OST3 ) τ ( α∈Γ Aα ) ≥ α∈Γ τ (Aα ), ∀{Aα } ⊂ 2X . The pair (X, τ ) is called an ordinary smooth topological space (in short, osts). We will denote the set of all ost’s on X as OST(X). Remark 2.2. Ying [8] called the mapping τ : 2X → I [resp. τ : I X → 2 and τ : I X → I] satisfying the axioms in Defi ...
Introduction to Combinatorial Homotopy Theory
... (homeomorphic to) a circle S 1 = {(x, y) ∈ R2 st x2 + y 2 = 1}. Three vertices and three edges are necessary: X = (V, S) with V = 2 = {0, 1, 2} and S = {0, 1, 2, 01, 02, 12} where as usual 01 is a shorthand for {0, 1}. ...
... (homeomorphic to) a circle S 1 = {(x, y) ∈ R2 st x2 + y 2 = 1}. Three vertices and three edges are necessary: X = (V, S) with V = 2 = {0, 1, 2} and S = {0, 1, 2, 01, 02, 12} where as usual 01 is a shorthand for {0, 1}. ...
Introduction to Topology
... (2) For each x ∈ X and each basis element B ∈ B containing x, there is a basis element B 0 ∈ B such that x ∈ B 0 ⊂ B. Proof. (2)⇒(1) Given U ∈ T , let x ∈ U. Since B generates T , there is B ∈ B such that x ∈ C ⊂ U. By hypothesis (2), there is B 0 ∈ B 0 such that x ∈ B 0 ⊂ B. Then x ∈ B 0 ⊂ U and by ...
... (2) For each x ∈ X and each basis element B ∈ B containing x, there is a basis element B 0 ∈ B such that x ∈ B 0 ⊂ B. Proof. (2)⇒(1) Given U ∈ T , let x ∈ U. Since B generates T , there is B ∈ B such that x ∈ C ⊂ U. By hypothesis (2), there is B 0 ∈ B 0 such that x ∈ B 0 ⊂ B. Then x ∈ B 0 ⊂ U and by ...
αAB-SETS IN IDEAL TOPOLOGICAL SPACES
... (2) ⇒(1): Let H be a pre--open and an α-set. By Proposition 2.13, H is semi-open. Since H is semi--open and pre-- open, by Proposition 1.6, it is an α--open. Example 2.18. (1) Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b}, X} and ={∅,{a}}.Then {b, c} is an αset but not a pre--open. ( ...
... (2) ⇒(1): Let H be a pre--open and an α-set. By Proposition 2.13, H is semi-open. Since H is semi--open and pre-- open, by Proposition 1.6, it is an α--open. Example 2.18. (1) Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b}, X} and ={∅,{a}}.Then {b, c} is an αset but not a pre--open. ( ...
on the ubiquity of simplicial objects
... where for any y ∈ K1 we have d0 y ∼ d1 y. We call π0 (K) the set of path-connected components of K, and K is said to be path-connected if π0 (K) contains only a single element. Proposition 2.2.1. Let (K, k0 ) be a Kan pair. Then πn (K, k0 ) is a group for n ≥ 1. Proof. Take α, β ∈ πn (K, k0 ). We de ...
... where for any y ∈ K1 we have d0 y ∼ d1 y. We call π0 (K) the set of path-connected components of K, and K is said to be path-connected if π0 (K) contains only a single element. Proposition 2.2.1. Let (K, k0 ) be a Kan pair. Then πn (K, k0 ) is a group for n ≥ 1. Proof. Take α, β ∈ πn (K, k0 ). We de ...
On the identification and establishment of topological
... et al., 1995). The great advantage of quantitative navigation methods is that with only one level in the spatial representation different tasks can be performed. For example, the information captured by the sensors can be used directly to build a representation of space, the method of local and glob ...
... et al., 1995). The great advantage of quantitative navigation methods is that with only one level in the spatial representation different tasks can be performed. For example, the information captured by the sensors can be used directly to build a representation of space, the method of local and glob ...
