Connected and hyperconnected generalized topological spaces 1
... such type are topological spaces defined by a closure operation. Generalized topological space is an important generalization of topological spaces. In the past decade, Csázar [4–10] and others have been considering generalized topological spaces, and developing a theory for them. more precisely, f ...
... such type are topological spaces defined by a closure operation. Generalized topological space is an important generalization of topological spaces. In the past decade, Csázar [4–10] and others have been considering generalized topological spaces, and developing a theory for them. more precisely, f ...
Metrisability of Manifolds in Terms of Function Spaces
... general may collapse in the presence of additional properties. These additional properties may be algebraic (e.g., a topological group is metrisable if and only if it is first-countable) or purely topological. For instance, a large collection of topological properties which are different in general ...
... general may collapse in the presence of additional properties. These additional properties may be algebraic (e.g., a topological group is metrisable if and only if it is first-countable) or purely topological. For instance, a large collection of topological properties which are different in general ...
Stratified fibre bundles
... F-bundles. (4.5) Example. By comparing definition (4.4) and definition (3.2) it is easy to see that a F-complex with the skeletal filtration is a F-stratified bundle since the spaces (V × Dn , V × S n−1 ) are trivial F-bundles over the CW-pair (Dn , S n−1 ). Moreover, if X is a F-stratified bundle s ...
... F-bundles. (4.5) Example. By comparing definition (4.4) and definition (3.2) it is easy to see that a F-complex with the skeletal filtration is a F-stratified bundle since the spaces (V × Dn , V × S n−1 ) are trivial F-bundles over the CW-pair (Dn , S n−1 ). Moreover, if X is a F-stratified bundle s ...
On Totally sg-Continuity, Strongly sg
... q. Then clearly f is totally sg-continuous, but not totally continuous. Definition 3.2 A function f : (X, τ) → (Y, σ) is said to be strongly semigeneralized-continuous (briefly strongly sg-continuous) if the inverse image of every subset of (Y, σ) is a sg-clopen subset of (X, τ). It is clear that st ...
... q. Then clearly f is totally sg-continuous, but not totally continuous. Definition 3.2 A function f : (X, τ) → (Y, σ) is said to be strongly semigeneralized-continuous (briefly strongly sg-continuous) if the inverse image of every subset of (Y, σ) is a sg-clopen subset of (X, τ). It is clear that st ...
RELATIONS BETWEEN UNION AND INTERSECTION OF IDEALS
... To prove the converse assume that for any collection {Iα } of ideals on (X, T ), TI = T0 where I = ∩Iα and T0 = ∩Tα . To prove (X, T ) is Alexandroff let {Uα }α∈Λ be a collection of open sets in (X, T ). Let U = ∩Uα . We claim that U is open in (X, T ). For all α ∈ Λ, let Aα = Uα −U and let Iα be the ...
... To prove the converse assume that for any collection {Iα } of ideals on (X, T ), TI = T0 where I = ∩Iα and T0 = ∩Tα . To prove (X, T ) is Alexandroff let {Uα }α∈Λ be a collection of open sets in (X, T ). Let U = ∩Uα . We claim that U is open in (X, T ). For all α ∈ Λ, let Aα = Uα −U and let Iα be the ...
On Iπgβ⋆ -closed sets in ideal topological spaces
... βI⋆ -open sets U, V of X such that A ⊆ U and B ⊆ V. Proposition 4.2. If X is a quasi-β-normal space, then X is quasi-β ⋆ -I-normal. Proof. It is obtained from Proposition 2.5. Theorem 4.3. The following properties are equivalent for a space X: (1) X is quasi-β ⋆ -I-normal, (2) for any disjoint π-clo ...
... βI⋆ -open sets U, V of X such that A ⊆ U and B ⊆ V. Proposition 4.2. If X is a quasi-β-normal space, then X is quasi-β ⋆ -I-normal. Proof. It is obtained from Proposition 2.5. Theorem 4.3. The following properties are equivalent for a space X: (1) X is quasi-β ⋆ -I-normal, (2) for any disjoint π-clo ...
Non-Hausdorff multifunction generalization of the Kelley
... 3.1, 3.2, τp on_F is jointly continuous. Since (Y ) 0 is a Tychonoff set, by Lemma 2.1, F is τp-compact. Then, by the Lemma 8 of Smithson [9, p. 258], F is equicontinuous. 4. Ascoli theorem. Let X = (X, r ) be a topological space. The k-extension of r is the family fc(τ) of all subsets U of X such t ...
... 3.1, 3.2, τp on_F is jointly continuous. Since (Y ) 0 is a Tychonoff set, by Lemma 2.1, F is τp-compact. Then, by the Lemma 8 of Smithson [9, p. 258], F is equicontinuous. 4. Ascoli theorem. Let X = (X, r ) be a topological space. The k-extension of r is the family fc(τ) of all subsets U of X such t ...
On Ψ~ e G-sets in grill topological spaces
... A ⊆ Int(Cl(A)) (resp. A ⊆ Int(Φ(A))). The collection of all peropen (resp. Φopen) sets in a topological space (X, τ ) is denoted by P O(X, τ ) (resp. ΦO(X, τ )). Definition 4. Let (X, τ, G) be a grill topological space. A grill G is said to be anti-codense grill if τ − {φ} ⊆ G. Theorem 5. [16] Let ( ...
... A ⊆ Int(Cl(A)) (resp. A ⊆ Int(Φ(A))). The collection of all peropen (resp. Φopen) sets in a topological space (X, τ ) is denoted by P O(X, τ ) (resp. ΦO(X, τ )). Definition 4. Let (X, τ, G) be a grill topological space. A grill G is said to be anti-codense grill if τ − {φ} ⊆ G. Theorem 5. [16] Let ( ...
Sufficient Conditions for Paracompactness of
... since its definition does not make it obvious how to verify that a space is paracompact, we give in these notes some conditions that suffice for paracompactness of a finitedimensional Hausdorff manifold. Henceforth in these notes, “manifold” always means “topological, finite-dimensional manifold”. W ...
... since its definition does not make it obvious how to verify that a space is paracompact, we give in these notes some conditions that suffice for paracompactness of a finitedimensional Hausdorff manifold. Henceforth in these notes, “manifold” always means “topological, finite-dimensional manifold”. W ...
Takashi Noiri and Valeriu Popa THE UNIFIED THEORY
... Definition 4.3. A function f : (X, mX ) → (Y, mY ) is said to be ωm-closed (resp. M-closed [15]) if f (A) is ω-m-closed (resp. mY -closed) in (Y, mY ) for every mX -closed set A of (X, mX ). Lemma 4.2. For a function f : (X, mX ) → (Y, mY ), the following properties are equivalent: (1) f is ω-m-clos ...
... Definition 4.3. A function f : (X, mX ) → (Y, mY ) is said to be ωm-closed (resp. M-closed [15]) if f (A) is ω-m-closed (resp. mY -closed) in (Y, mY ) for every mX -closed set A of (X, mX ). Lemma 4.2. For a function f : (X, mX ) → (Y, mY ), the following properties are equivalent: (1) f is ω-m-clos ...