Higson have introduced the conception of "Higson’s corona" (see [1]). For a given metric space X, it is a kind of compactification of X related to the metric d on it. Denote by BR(X) the set {y ∈ X\d(x,y) ...Higson have introduced the conception of "Higson’s corona" (see [1]). For a given metric space X, it is a kind of compactification of X related to the metric d on it. Denote by BR(X) the set {y ∈ X\d(x,y) < R}. Recall that a slowly oscillating function on X is a function f G C*(X) satisfying the following condition:展开更多
By means of a characterization of compact spaces in terms of open CD*-filters induced by a , a - and open CD*-filters process of compactifications of an arbitrary topological space Y is obtained in Sec. 3 by embedding...By means of a characterization of compact spaces in terms of open CD*-filters induced by a , a - and open CD*-filters process of compactifications of an arbitrary topological space Y is obtained in Sec. 3 by embedding Y as a dense subspace of , YS = {ε |ε is an open CD*-filter that does not converge in Y}, YT = {A|A is a basic open CD*-filter that does not converge in Y}, is the topology induced by the base B = {U*|U is open in Y, U ≠φ} and U* = {F∈Ysw (or YTw)|U∈F}. Furthermore, an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X?can be obtained from a by the?similar process in Sec.3.展开更多
Closed and basic closed C*D-filters are used in a process similar to Wallman method for compactifications of the topological spaces Y, of which, there is a subset of C*(Y) containing a non-constant function, where C*(...Closed and basic closed C*D-filters are used in a process similar to Wallman method for compactifications of the topological spaces Y, of which, there is a subset of C*(Y) containing a non-constant function, where C*(Y) is the set of bounded real continuous functions on Y. An arbitrary Hausdorff compactification (Z,h) of a Tychonoff space X can be obtained by using basic closed C*D-filters from in a similar way, where C(Z) is the set of real continuous functions on Z.展开更多
Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the se...Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1^＊, x2^＊ …} are investigated. Various estimates of the defect d（T） = codim R（T）, where R（T） is the range of T, are given. The case of xn^＊ = dnxtn^＊,where dn ∈ R and xtn^＊ 〉 0 are extreme points of the unit ball Be∞^＊, that is, tn ∈ βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R（T） are given and the value d（T） is calculated. For example, the condition {n ： 0 〈 ｜da｜ 〈 δ}= θ for some 5 is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R（T） （tn0 is isolated if there is a neighborhood U of tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, then d（T） is equal to the defect δ{tn} of {tn}. It is shown that if d（T） = ∞ and R（T） is closed, then there exists a sequence {An} of pairwise disjoint subsets of N satisfying XAn ∈ R（T）.展开更多
In 1977,Yingming Liu introduced quasi-paracompactness and proved that under 2^(ω1)>2~ωevery separable normal quasi-paracompact space is a paracompact space,which is a result of set-theoretic topology.In this pape...In 1977,Yingming Liu introduced quasi-paracompactness and proved that under 2^(ω1)>2~ωevery separable normal quasi-paracompact space is a paracompact space,which is a result of set-theoretic topology.In this paper we further prove that hypothesis“2^(ω1)>2~ω”is equivalent to that every separable normal quasi-paracompact space is a paracompact space,which gives an independent result of quasi-paracompactness.展开更多
In the spirit of Morse homology initiated by Witten and Floer,we construct two∞-categories A and B.The weak one A comes out of the Morse-Smale pairs and their higher homotopies,and the strict one B concerns the chain...In the spirit of Morse homology initiated by Witten and Floer,we construct two∞-categories A and B.The weak one A comes out of the Morse-Smale pairs and their higher homotopies,and the strict one B concerns the chain complexes of the Morse functions.Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters,we build up a weak∞-functor F:A→B.Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.展开更多
文摘Higson have introduced the conception of "Higson’s corona" (see [1]). For a given metric space X, it is a kind of compactification of X related to the metric d on it. Denote by BR(X) the set {y ∈ X\d(x,y) < R}. Recall that a slowly oscillating function on X is a function f G C*(X) satisfying the following condition:
文摘By means of a characterization of compact spaces in terms of open CD*-filters induced by a , a - and open CD*-filters process of compactifications of an arbitrary topological space Y is obtained in Sec. 3 by embedding Y as a dense subspace of , YS = {ε |ε is an open CD*-filter that does not converge in Y}, YT = {A|A is a basic open CD*-filter that does not converge in Y}, is the topology induced by the base B = {U*|U is open in Y, U ≠φ} and U* = {F∈Ysw (or YTw)|U∈F}. Furthermore, an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X?can be obtained from a by the?similar process in Sec.3.
文摘Closed and basic closed C*D-filters are used in a process similar to Wallman method for compactifications of the topological spaces Y, of which, there is a subset of C*(Y) containing a non-constant function, where C*(Y) is the set of bounded real continuous functions on Y. An arbitrary Hausdorff compactification (Z,h) of a Tychonoff space X can be obtained by using basic closed C*D-filters from in a similar way, where C(Z) is the set of real continuous functions on Z.
文摘Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1^＊, x2^＊ …} are investigated. Various estimates of the defect d（T） = codim R（T）, where R（T） is the range of T, are given. The case of xn^＊ = dnxtn^＊,where dn ∈ R and xtn^＊ 〉 0 are extreme points of the unit ball Be∞^＊, that is, tn ∈ βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R（T） are given and the value d（T） is calculated. For example, the condition {n ： 0 〈 ｜da｜ 〈 δ}= θ for some 5 is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R（T） （tn0 is isolated if there is a neighborhood U of tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, then d（T） is equal to the defect δ{tn} of {tn}. It is shown that if d（T） = ∞ and R（T） is closed, then there exists a sequence {An} of pairwise disjoint subsets of N satisfying XAn ∈ R（T）.
基金the National Natural Science Foundation of China (Grant No. 12171015)the Natural Science Foundation of Fujian Province (Grant Nos. 2020J014282020J05230)
文摘In 1977,Yingming Liu introduced quasi-paracompactness and proved that under 2^(ω1)>2~ωevery separable normal quasi-paracompact space is a paracompact space,which is a result of set-theoretic topology.In this paper we further prove that hypothesis“2^(ω1)>2~ω”is equivalent to that every separable normal quasi-paracompact space is a paracompact space,which gives an independent result of quasi-paracompactness.
基金Supported by National Key R&D Program of China(Grant No.2020YFA0713300)NSFC(Grant Nos.11771303,12171327,11911530092,11871045)。
文摘In the spirit of Morse homology initiated by Witten and Floer,we construct two∞-categories A and B.The weak one A comes out of the Morse-Smale pairs and their higher homotopies,and the strict one B concerns the chain complexes of the Morse functions.Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters,we build up a weak∞-functor F:A→B.Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.
