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
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).