In this paper, some generalized metric properties in strongly topological gyrogroups are studied.... more In this paper, some generalized metric properties in strongly topological gyrogroups are studied. In particular, it is proved that when G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a second-countable admissible subgyrogroup generated from U , if the quotient space G/H is an ℵ0-space (resp., cosmic space), then G is also an ℵ0-space (resp., cosmic space); If the quotient space G/H has a star-countable cs-network (resp., wcs *-network, k-network), then G also has a star-countable cs-network (resp., wcs *-network, k-network). Moreover, it is shown that when G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a locally compact metrizable admissible subgyrogroup generated from U , if the quotient space G/H is sequential, then G is also sequential; Furthermore, if the quotient space G/H is strictly (strongly) Fréchet-Urysohn, then G is also strictly (
We prove that (1) for any complete lattice L, the set D(L) of all nonempty saturated compact subs... more We prove that (1) for any complete lattice L, the set D(L) of all nonempty saturated compact subsets of the Scott space of L is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice L is non-sober, then the Scott space of D(L) is non-sober. Using these results and the Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus give a positive answer to a problem posed by Jung. We will also prove that a T 0 space is well-filtered iff its upper space (the set D(X) of all nonempty saturated compact subsets of X equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.
Based on topological Rudin's Lemma, we investigate two new kinds of sets-Rudin sets and well-filt... more Based on topological Rudin's Lemma, we investigate two new kinds of sets-Rudin sets and well-filtered determined sets in T 0 topological spaces. Using such sets, we formulate and prove some new characterizations for well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on wellfiltered spaces. Our study also lead to a new class of spaces-strong d-spaces and some problems whose solutions will strengthen our understanding of the related structures.
For a T 0 space X, let K(X) be the poset of all compact saturated sets of X with the reverse incl... more For a T 0 space X, let K(X) be the poset of all compact saturated sets of X with the reverse inclusion order. The space X is said to have property Q if for any K 1 , K 2 ∈ K(X), K 2 ≪ K 1 in K(X) iff K 2 ⊆ int K 1. In this paper, we give several connections among the well-filteredness of X, the sobriety of X, the local compactness of X, the core compactness of X, the property Q of X, the coincidence of the upper Vietoris topology and Scott topology on K(X), and the continuity of x → ↑x : X −→ ΣK(X) (where ΣK(X) is the Scott space of K(X)). It is shown that for a well-filtered space X for which its Smyth power space P S (X) is first-countable, the following three properties are equivalent: the local compactness of X, the core compactness of X and the continuity of K(X). It is also proved that for a first-countable T 0 space X in which the set of minimal elements of K is countable for any compact saturated subset K of X, the Smyth power space P S (X) is first-countable. For the Alexandroff double circle Y , which is Hausdorff and first-countable, we show that its Smyth power space P S (Y) is not first-countable.
For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets o... more For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$ . $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$ ) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$ . A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first...
In this paper,the concepts of semi-Smooth lattices and semi-Smooth algebraic lattices are introdu... more In this paper,the concepts of semi-Smooth lattices and semi-Smooth algebraic lattices are introduced.Some basic properties of them and the relation with semi-continuous lattices are investigated.It is proved that a complete lattice is completely distributive lattice if and only if it is both semi-Smooth lattice and semi-continuous lattice.
In this paper, we mainly investigate the conditions under which the Scott topology on the product... more In this paper, we mainly investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Some such conditions are given.
Topology - Recent Advances and Applications [Working Title]
In most topology books, the Hausdorff separation property is assumed from the very start and cont... more In most topology books, the Hausdorff separation property is assumed from the very start and contain very little information on non-Hausdorff spaces. In classical mathematics, most topological spaces are indeed Hausdorff. But non-Hausdorff spaces are important already in algebraic geometry, and crucial in fields such as domain theory. Indeed, in connection with order, non-Hausdorff spaces, especially $T_0$ spaces, play a more significant role than Hausdorff spaces. Sobriety is probably the most important and useful property of non-Hausdorff topological spaces. It has been used in the characterizations of spectral spaces and $T_0$ spaces that are determined by their open set lattices. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology theory: $d$-space and well-filtered space. In the past few years, some remarkable progresses have been achieved in understanding such structures. In this chap...
The author gives some characterizations of generalized λ hypercontinuous lattices. In particular,... more The author gives some characterizations of generalized λ hypercontinuous lattices. In particular, we prove that the λ-interval topology θ_λ(L) on a complete lattice L is strictly T_2 L is generalized λ hypercontinuous the relation on L is generalized λ regular.
The concept of property MF relative to a base of quasicontinuous domain is introduced.It is prove... more The concept of property MF relative to a base of quasicontinuous domain is introduced.It is proved that for a quasicontinuous domain P,the intersection of any two Scott upper compact sets is still Scott compact if and only if P satisfies property MF with respect to any(some) base.
The theory of continuous domains, due to its strong background in computer science, general topol... more The theory of continuous domains, due to its strong background in computer science, general topology and logic has been extensively studied by people from various areas (see Abramsky and Jung 1994; Gierz et al. 1980, 2003). Since many models may not be dcpos, an important direction in the study of continuous domains is to extend the theory of continuous domains to that of posets as much as possible (
In the paper,some mapping properties of semicontinuous lattices and semialgebraic lattices are in... more In the paper,some mapping properties of semicontinuous lattices and semialgebraic lattices are investigated,the function spaces of strongly continuous lattices are discussed,and an embedding theorem for strongly continuous lattices into cubes is given.
In this paper the classical theorem of Zareckii about regular relations is generalized and an int... more In this paper the classical theorem of Zareckii about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckii theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the UrysohnNachbin lemma is presented which is quite different from the classical one.
In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, ... more In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. The concepts of irreducible subset systems (R-subset systems for short), H-sober spaces and super H-sober spaces for a general R-subset system H are introduced. It is proved that the product space of a family of T 0 spaces is H-sober iff each factor space is H-sober, and if H has a natural property (called property M), then the super H-sobriety is a special type of H-sobriety, and hence the product space of a family of T 0 spaces is super H-sober iff each factor space is super Hsober. Let Top 0 be the category of all T 0 spaces with continuous mappings. For a T 0 space X and an H-sober space Y , we show that the function space Top 0 (X, Y) equipped with the topology of pointwise convergence is H-sober. Furthermore, if H has property M and Y is a super H-sober space, then the function space Top 0 (X, Y) equipped with the topology of pointwise convergence is super H-sober. One immediate corollary is that for a T 0 space X and a well-filtered space Y , the function space Top 0 (X, Y) equipped with the topology of pointwise convergence is well-filtered. For an R-subset system H having property M, the Smyth power space of a H-sober space is not H-sober in general. But for the super H-sobriety, we prove that a T 0 space X is super H-sober iff its Smyth power space P S (X) is super H-sober. A direct construction of the H-sobrifcations and super H-sobrifications of T 0 spaces is given. So the category of all H-sober spaces is reflective in Top 0 , and the category of all super H-sober spaces is also reflective in Top 0 if H has property M. It is shown that the H-sobrification preserves finite products of T 0 spaces, and the super H-sobrification preserves finite products of T 0 spaces if H has property M.
In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible ... more In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mat...
In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such... more In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $ k $-bounded sober spaces and weakly sober spaces. For $ T_{0} $ spaces $ X $ and $ Y $, it is proved that $ Y $ is bounded sober iff the function space $ {\bf{Top}}(X, Y) $ of all continuous functions $ f : X\longrightarrow Y $ equipped with the pointwise convergence topology is bounded sober iff $ {\bf{Top}}(X, Y) $ equipped with the Isbell topology is bounded sober. But for a $ k $-bounded sober space $ X $, the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology may not be $ k $-bounded sober. It is shown that if the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $ Y $ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.
In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyr... more In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyrogroup and H is a strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is first-countable if and only if G/H is a bisequential space if and only if G/H is a weakly first-countable space if and only if G/H is a csf-countable and sequential α 7-space. Moreover, it is shown that if G is a strongly topological gyrogroup and H is a locally compact strong subgyrogroup of G, then there exists an open neighborhood U of the identity element 0 such that π(U) is closed in G/H and the restriction of π to U is a perfect mapping from U onto the subspace π(U); if H is a locally compact metrizable strong subgyrogroup of G and the quotient space G/H is sequential, then G is also sequential; if H is a closed first-countable and separable strong subgyrogroup of G, the quotient space G/H is an ℵ 0-space, then G is an ℵ 0-space; if the quotient space G/H is a cosmic space, then G is also a cosmic space; if the quotient space G/H has a star-countable cs-network or star-countable wcs *-network, then G also has a star-countable cs-network or star-countable wcs *-network, respectively.
Based on the concepts of R-factorizable topological groups and M-factorizable topological groups,... more Based on the concepts of R-factorizable topological groups and M-factorizable topological groups, we introduce four classes of factorizabilities on topological groups, named P M-factorizabilities, P m-factorizabilities, SM-factorizabilities and P SM-factorizabilities, respectively. Some properties of the four classes of spaces are investigated.
In this paper, some generalized metric properties in strongly topological gyrogroups are studied.... more In this paper, some generalized metric properties in strongly topological gyrogroups are studied. In particular, it is proved that when G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a second-countable admissible subgyrogroup generated from U , if the quotient space G/H is an ℵ0-space (resp., cosmic space), then G is also an ℵ0-space (resp., cosmic space); If the quotient space G/H has a star-countable cs-network (resp., wcs *-network, k-network), then G also has a star-countable cs-network (resp., wcs *-network, k-network). Moreover, it is shown that when G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a locally compact metrizable admissible subgyrogroup generated from U , if the quotient space G/H is sequential, then G is also sequential; Furthermore, if the quotient space G/H is strictly (strongly) Fréchet-Urysohn, then G is also strictly (
We prove that (1) for any complete lattice L, the set D(L) of all nonempty saturated compact subs... more We prove that (1) for any complete lattice L, the set D(L) of all nonempty saturated compact subsets of the Scott space of L is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice L is non-sober, then the Scott space of D(L) is non-sober. Using these results and the Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus give a positive answer to a problem posed by Jung. We will also prove that a T 0 space is well-filtered iff its upper space (the set D(X) of all nonempty saturated compact subsets of X equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.
Based on topological Rudin's Lemma, we investigate two new kinds of sets-Rudin sets and well-filt... more Based on topological Rudin's Lemma, we investigate two new kinds of sets-Rudin sets and well-filtered determined sets in T 0 topological spaces. Using such sets, we formulate and prove some new characterizations for well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on wellfiltered spaces. Our study also lead to a new class of spaces-strong d-spaces and some problems whose solutions will strengthen our understanding of the related structures.
For a T 0 space X, let K(X) be the poset of all compact saturated sets of X with the reverse incl... more For a T 0 space X, let K(X) be the poset of all compact saturated sets of X with the reverse inclusion order. The space X is said to have property Q if for any K 1 , K 2 ∈ K(X), K 2 ≪ K 1 in K(X) iff K 2 ⊆ int K 1. In this paper, we give several connections among the well-filteredness of X, the sobriety of X, the local compactness of X, the core compactness of X, the property Q of X, the coincidence of the upper Vietoris topology and Scott topology on K(X), and the continuity of x → ↑x : X −→ ΣK(X) (where ΣK(X) is the Scott space of K(X)). It is shown that for a well-filtered space X for which its Smyth power space P S (X) is first-countable, the following three properties are equivalent: the local compactness of X, the core compactness of X and the continuity of K(X). It is also proved that for a first-countable T 0 space X in which the set of minimal elements of K is countable for any compact saturated subset K of X, the Smyth power space P S (X) is first-countable. For the Alexandroff double circle Y , which is Hausdorff and first-countable, we show that its Smyth power space P S (Y) is not first-countable.
For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets o... more For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$ . $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$ ) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$ . A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first...
In this paper,the concepts of semi-Smooth lattices and semi-Smooth algebraic lattices are introdu... more In this paper,the concepts of semi-Smooth lattices and semi-Smooth algebraic lattices are introduced.Some basic properties of them and the relation with semi-continuous lattices are investigated.It is proved that a complete lattice is completely distributive lattice if and only if it is both semi-Smooth lattice and semi-continuous lattice.
In this paper, we mainly investigate the conditions under which the Scott topology on the product... more In this paper, we mainly investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Some such conditions are given.
Topology - Recent Advances and Applications [Working Title]
In most topology books, the Hausdorff separation property is assumed from the very start and cont... more In most topology books, the Hausdorff separation property is assumed from the very start and contain very little information on non-Hausdorff spaces. In classical mathematics, most topological spaces are indeed Hausdorff. But non-Hausdorff spaces are important already in algebraic geometry, and crucial in fields such as domain theory. Indeed, in connection with order, non-Hausdorff spaces, especially $T_0$ spaces, play a more significant role than Hausdorff spaces. Sobriety is probably the most important and useful property of non-Hausdorff topological spaces. It has been used in the characterizations of spectral spaces and $T_0$ spaces that are determined by their open set lattices. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology theory: $d$-space and well-filtered space. In the past few years, some remarkable progresses have been achieved in understanding such structures. In this chap...
The author gives some characterizations of generalized λ hypercontinuous lattices. In particular,... more The author gives some characterizations of generalized λ hypercontinuous lattices. In particular, we prove that the λ-interval topology θ_λ(L) on a complete lattice L is strictly T_2 L is generalized λ hypercontinuous the relation on L is generalized λ regular.
The concept of property MF relative to a base of quasicontinuous domain is introduced.It is prove... more The concept of property MF relative to a base of quasicontinuous domain is introduced.It is proved that for a quasicontinuous domain P,the intersection of any two Scott upper compact sets is still Scott compact if and only if P satisfies property MF with respect to any(some) base.
The theory of continuous domains, due to its strong background in computer science, general topol... more The theory of continuous domains, due to its strong background in computer science, general topology and logic has been extensively studied by people from various areas (see Abramsky and Jung 1994; Gierz et al. 1980, 2003). Since many models may not be dcpos, an important direction in the study of continuous domains is to extend the theory of continuous domains to that of posets as much as possible (
In the paper,some mapping properties of semicontinuous lattices and semialgebraic lattices are in... more In the paper,some mapping properties of semicontinuous lattices and semialgebraic lattices are investigated,the function spaces of strongly continuous lattices are discussed,and an embedding theorem for strongly continuous lattices into cubes is given.
In this paper the classical theorem of Zareckii about regular relations is generalized and an int... more In this paper the classical theorem of Zareckii about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckii theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the UrysohnNachbin lemma is presented which is quite different from the classical one.
In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, ... more In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. The concepts of irreducible subset systems (R-subset systems for short), H-sober spaces and super H-sober spaces for a general R-subset system H are introduced. It is proved that the product space of a family of T 0 spaces is H-sober iff each factor space is H-sober, and if H has a natural property (called property M), then the super H-sobriety is a special type of H-sobriety, and hence the product space of a family of T 0 spaces is super H-sober iff each factor space is super Hsober. Let Top 0 be the category of all T 0 spaces with continuous mappings. For a T 0 space X and an H-sober space Y , we show that the function space Top 0 (X, Y) equipped with the topology of pointwise convergence is H-sober. Furthermore, if H has property M and Y is a super H-sober space, then the function space Top 0 (X, Y) equipped with the topology of pointwise convergence is super H-sober. One immediate corollary is that for a T 0 space X and a well-filtered space Y , the function space Top 0 (X, Y) equipped with the topology of pointwise convergence is well-filtered. For an R-subset system H having property M, the Smyth power space of a H-sober space is not H-sober in general. But for the super H-sobriety, we prove that a T 0 space X is super H-sober iff its Smyth power space P S (X) is super H-sober. A direct construction of the H-sobrifcations and super H-sobrifications of T 0 spaces is given. So the category of all H-sober spaces is reflective in Top 0 , and the category of all super H-sober spaces is also reflective in Top 0 if H has property M. It is shown that the H-sobrification preserves finite products of T 0 spaces, and the super H-sobrification preserves finite products of T 0 spaces if H has property M.
In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible ... more In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mat...
In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such... more In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $ k $-bounded sober spaces and weakly sober spaces. For $ T_{0} $ spaces $ X $ and $ Y $, it is proved that $ Y $ is bounded sober iff the function space $ {\bf{Top}}(X, Y) $ of all continuous functions $ f : X\longrightarrow Y $ equipped with the pointwise convergence topology is bounded sober iff $ {\bf{Top}}(X, Y) $ equipped with the Isbell topology is bounded sober. But for a $ k $-bounded sober space $ X $, the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology may not be $ k $-bounded sober. It is shown that if the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $ Y $ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.
In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyr... more In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyrogroup and H is a strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is first-countable if and only if G/H is a bisequential space if and only if G/H is a weakly first-countable space if and only if G/H is a csf-countable and sequential α 7-space. Moreover, it is shown that if G is a strongly topological gyrogroup and H is a locally compact strong subgyrogroup of G, then there exists an open neighborhood U of the identity element 0 such that π(U) is closed in G/H and the restriction of π to U is a perfect mapping from U onto the subspace π(U); if H is a locally compact metrizable strong subgyrogroup of G and the quotient space G/H is sequential, then G is also sequential; if H is a closed first-countable and separable strong subgyrogroup of G, the quotient space G/H is an ℵ 0-space, then G is an ℵ 0-space; if the quotient space G/H is a cosmic space, then G is also a cosmic space; if the quotient space G/H has a star-countable cs-network or star-countable wcs *-network, then G also has a star-countable cs-network or star-countable wcs *-network, respectively.
Based on the concepts of R-factorizable topological groups and M-factorizable topological groups,... more Based on the concepts of R-factorizable topological groups and M-factorizable topological groups, we introduce four classes of factorizabilities on topological groups, named P M-factorizabilities, P m-factorizabilities, SM-factorizabilities and P SM-factorizabilities, respectively. Some properties of the four classes of spaces are investigated.
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