共查询到20条相似文献,搜索用时 31 毫秒
1.
A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset AX the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the Gδ-measurability (the preimage of each open set is of type Gδ). Also under Martin Axiom, we construct a Gδ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov. 相似文献
2.
Jie-Hua Mai 《Topology and its Applications》2011,158(16):2216-2220
Let X be a topological space, f:X→X be a continuous map, and Y be a compact, connected and closed subset of X. In this paper we show that, if the boundary X∂Y contains exactly one point v and f(v)∈Y, then Y contains a minimal set of f. 相似文献
3.
Martin Čadek Marek Krčál Jiří Matoušek Lukáš Vokřínek Uli Wagner 《Discrete and Computational Geometry》2014,51(1):24-66
We consider two basic problems of algebraic topology: the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity. The extension problem is the following: Given topological spaces X and Y, a subspace A?X, and a (continuous) map f:A→Y, decide whether f can be extended to a continuous map $\bar{f}\colon X\to Y$ . All spaces are given as finite simplicial complexes, and the map f is simplicial. Recent positive algorithmic results, proved in a series of companion papers, show that for (k?1)-connected Y, k≥2, the extension problem is algorithmically solvable if the dimension of X is at most 2k?1, and even in polynomial time when k is fixed. Here we show that the condition $\mathop{\mathrm{dim}}\nolimits X\leq 2k-1$ cannot be relaxed: for $\mathop{\mathrm{dim}}\nolimits X=2k$ , the extension problem with (k?1)-connected Y becomes undecidable. Moreover, either the target space Y or the pair (X,A) can be fixed in such a way that the problem remains undecidable. Our second result, a strengthening of a result of Anick, says that the computation of π k (Y) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input. 相似文献
4.
We introduce an alternative definition of fibrewise uniformity and discuss consequences deduced from new axioms. By modifying James’ definition of fibrewise uniform structure, which is a slightly strengthened one, we define a new fibrewise uniformity which is symmetric in global and realizes 1-1 correspondence between fibrewise entourage uniformities and fibrewise covering uniformities. Moreover, we obtain a characterization of the fibrewise completion of fibrewise generalized uniform space as a fibrewise extension of a fibrewise space. As an application of the fibrewise completion theory, we show that there exists a fibrewise Shanin compactification of a fibrewise space. Finally, we study extendability of fibrewise maps from dense subspaces. That is, for a fibrewise space X, A ? X dense in X and a fibrewise continuous map f: A → Y, when can f be extended to the whole space X? Many characterization theorems of extendable fibrewise continuous maps are given. 相似文献
5.
A. A. Zaitov 《Mathematical Notes》2010,88(5-6):655-660
We establish that if X and Y are metric compacta and f: X → Y is a continuous surjective mapping, then the openness of the mapping OH(f): OH(X) → OH(Y) of spaces of weakly additive homogeneous functionals is equivalent to the openness of f. 相似文献
6.
A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f(\bar G) \subseteq \overline {f(G)} $ for any open connected set G ? X. We call a function f: X → Y segmentary connected if X is topological vector space and f([a, b]) is connected for every segment [a, b] ? X. We show that if X is a hereditarily Baire space, Y is a metric space, f: X → Y is a Baire-one function and one of the following conditions holds:
- X is a connected and locally connected space and f is a weakly Gibson function
- X is an arcwise connected space and f is a Darboux function
- X is a topological vector space and f is a segmentary connected function, then f has a connected graph.
7.
Alireza Kamel Mirmostafaee 《Mathematica Slovaca》2014,64(4):1019-1026
By means of a topological game, a class of topological spaces which contains compact spaces, q-spaces and W-spaces was defined in [BOUZIAD, A.: The Ellis theorem and continuity in groups, Topology Appl. 50 (1993), 73–80]. We will show that if Y belongs to this class, every separately continuous function f: X × Y → Z is jointly continuous on a dense subset of X × Y provided that X is σ-β-unfavorable and Z is a regular weakly developable space. 相似文献
8.
J. Ka?kol 《Journal of Mathematical Analysis and Applications》2003,280(1):155-162
It is well known that the space Cp([0,1]) has countable tightness but it is not Fréchet-Urysohn. Let X be a Cech-complete topological space. We prove that the space Cp(X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet-Urysohn if and only if Cp(X) has countable bounded tightness, i.e., for every subset A of Cp(X) and every x in the closure of A in Cp(X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided. 相似文献
9.
Tomás Domínguez Benavides 《Journal of Mathematical Analysis and Applications》1985,105(1):176-186
Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, a complete metric space formed by all α-nonexpansive mappings fC → A and a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in (2)the subset of formed by all α-contractive mappings is of Baire first category in ; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in . Some applications to the fixed point theory and calculation of the topological degree are given. 相似文献
10.
Harald Brandenburg 《Topology and its Applications》1985,20(1):17-27
Following Pareek a topological space X is called D-paracompact if for every open cover of X there exists a continuous mapping f from X onto a developable T1-space Y and an open cover of Y such that { f-1[B]|B ∈ } refines . It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces. 相似文献
11.
Haruto Ohta 《Topology and its Applications》1984,17(3):265-274
In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω1, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact. 相似文献
12.
We call a value y = f(x) of a map f: X → Y dimensionally regular if dimX ≤ dim(Y × f ?1(y)). It was shown in [6] that if a map f: X → Y between compact metric spaces does not have dimensionally regular values, then X is a Boltyanskii compactum, i.e., a compactum satisfying the equality dim(X × X) = 2dim X ? 1. In this paper we prove that every Boltyanskii compactum X of dimension dim X ≥ 6 admits a map f: X → Y without dimensionally regular values. We show that the converse does not hold by constructing a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value. 相似文献
13.
Jean Saint Raymond 《Topology and its Applications》2007,154(8):1714-1725
The aim of this note is to prove the following result: “Assume that X is a metric Borel space of class ξ, that is continuous, that every fiber f−1(y) is complete and that every countable compact subset of Y is the image by f of some compact subset of X. Then Y is Borel and moreover of class ξ”. We give also an extension to the case where the fibers are only assumed to be Polish. 相似文献
14.
M.G. Tkačenko 《Topology and its Applications》1983,15(1):93-98
We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Πα?AXα has a countable o-tightness and Y is a subset of X such that πB(Y) = Πα?BXα for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding. 相似文献
15.
Petr Holický 《Topology and its Applications》2010,157(12):1926-275
We show that a metrizable space Y is completely metrizable if there is a continuous surjection f:X→Y such that the images of open (clopen) subsets of the (0-dimensional paracompact) ?ech-complete space X are resolvable subsets of Y (in particular, e.g., the elements of the smallest algebra generated by open sets in Y). 相似文献
16.
John Wolfe 《Israel Journal of Mathematics》1974,18(2):133-140
A Banach spaceX is aP λ-space if wheneverX is isometrically embedded in another Banach spaceY there is a projection ofY ontoX with norm at most λ.C(T) denotes the Banach space of continuous real-valued functions on the compact Hausdorff spaceT. T satisfies the countable chain condition (CCC) if every family of disjoint non-empty open sets inT is countable.T is extremally disconnected if the closure of every open set inT is open. The main result is that ifT satisfies the CCC andC(T) is aP λ-space, thenT is the union of an open dense extremally disconnected subset and a complementary closed setT Asuch thatC(TA) is aP λ?1-space. 相似文献
17.
Roman Ger 《Results in Mathematics》1994,26(3-4):281-289
Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A o ∈ Y, Ak(x) ? Ak(x,…,x), x ∈ X, and the maps A k: X k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed. 相似文献
18.
M. Kulczycki 《Regular and Chaotic Dynamics》2008,13(2):81-84
Vu Dong Tô has proven in [1] that for any mapping f: X → X, where X is a metric space that is not precompact, the third condition in the Devaney’s definition of chaos follows from the first two even if f is not assumed to be continuous. This paper completes this result by analysing the precompact case. We show that if X is either finite or perfect one can always find a map f: X → X that satisfies the first two conditions of Devaney’s chaos but not the third. Additionally, if X is neither finite nor perfect there is no f: X → X that would satisfy the first two conditions of Devaney’s chaos at the same time. 相似文献
19.
Ondřej F.K. Kalenda 《Quaestiones Mathematicae》2016,39(3):351-362
Let ? : X → Y be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on Y in the space of real-valued affine continuous functions on X is complemented. We show that if F is a topological vector space, then f : Y → F is of affine Baire class α whenever the composition f ? ? is of affine Baire class α. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings. 相似文献
20.
J Globevnik 《Journal of Functional Analysis》1976,22(1):32-38
Denote by Δ(resp. ) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on , analytic on Δ and satisfying sup. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X. 相似文献