共查询到20条相似文献,搜索用时 15 毫秒
1.
M.R. Koushesh 《Topology and its Applications》2007,154(3):698-721
For a non-compact metrizable space X, let E(X) be the set of all one-point metrizable extensions of X, and when X is locally compact, let EK(X) denote the set of all locally compact elements of E(X) and be the order-anti-isomorphism (onto its image) defined in [M. Henriksen, L. Janos, R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolin. 46 (2005) 105-123; in short HJW]. By definition λ(Y)=?n<ωclβX(Un∩X)\X, where Y=X∪{p}∈E(X) and {Un}n<ω is an open base at p in Y. We characterize the elements of the image of λ as exactly those non-empty zero-sets of βX which miss X, and the elements of the image of EK(X) under λ, as those which are moreover clopen in βX\X. This answers a question of [HJW]. We then study the relation between E(X) and EK(X) and their order structures, and introduce a subset ES(X) of E(X). We conclude with some theorems on the cardinality of the sets E(X) and EK(X), and some open questions. 相似文献
2.
We consider various possiblities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type exetension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending separated continous functions in such a way that, certain continuous extensions remain separated. Functgions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3. 相似文献
3.
M.R. Koushesh 《Topology and its Applications》2007,154(14):2607-2634
If a Tychonoff space X is dense in a Tychonoff space Y, then Y is called a Tychonoff extension of X. Two Tychonoff extensions Y1 and Y2 of X are said to be equivalent, if there exists a homeomorphism which keeps X pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of X. We identify those extensions of X which belong to the same equivalence classes. For two Tychonoff extensions Y1 and Y2 of X, we write Y2?Y1, if there exists a continuous function which keeps X pointwise fixed. This is a partial order on the set of all (equivalence classes of) Tychonoff extensions of X. If a Tychonoff extension Y of X is such that Y\X is a singleton, then Y is called a one-point extension of X. Let T(X) denote the set of all one-point extensions of X. Our purpose is to study the order structure of the partially ordered set (T(X),?). For a locally compact space X, we define an order-anti-isomorphism from T(X) onto the set of all nonempty closed subsets of βX\X. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of X, the set of all one-point Lindelöf extensions of X, the set of all one-point pseudocompact extensions of X, and the set of all one-point ?ech-complete extensions of X, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of βX\X. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of C∗(X), partially ordered with inclusion, and the topology of subspaces of βX\X. We leave some problems open. 相似文献
4.
M.R. Koushesh 《Topology and its Applications》2011,158(3):509-532
A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y′ of X let Y?Y′ if there is a continuous function of Y′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.
Conjecture.
For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic. 相似文献
5.
Extensions for Sobolev mappings between manifolds 总被引:1,自引:0,他引:1
We consider two compact Riemannian manifoldsM andN, such thatM has a boundary (but notN). We address the extension problem in the Sobolev class, namely, we investigate the question: foru W
1–1/p,pM,N is there a mapV inW
1/p(M,N) such thatV=u on M. Various results are given, and an emphasis is put on the special (simple) caseN=S
1. 相似文献
6.
Dmitri Shakhmatov 《Topology and its Applications》2009,156(7):1216-1223
If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory. 相似文献
7.
Lajos Soukup 《Topology and its Applications》2008,155(4):347-353
Nagata conjectured that every M-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. Although this conjecture was refuted by Burke and van Douwen, and A. Kato, independently, but we can show that there is a c.c.c. poset P of size ω2 such that in VP Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space X∈V is an M-space in VP then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in VP). By a result of Morita, it is enough to show that every first countable regular space from the ground model has a first countable countably compact extension in VP. As a corollary, we also obtain that every first countable regular space from the ground model has a maximal first countable extension in model VP. 相似文献
8.
We describe the asymptotic behavior of automorphisms of totally disconnected locally compact groups in terms of a set of `directions'
which comes equipped with a natural pseudo-metric. The structure at infinity obtained by completing the induced metric quotient
space of the set of directions recovers familiar objects such as: the set of ends of the tree for the group of inner automorphisms
of the group of isometries of a regular locally finite tree; and the spherical Bruhat-Tits building for the group of inner
automorphisms of the set of rational points of a semisimple group over a local field.
Research supported by A.R.C. Grant DP0208137. 相似文献
9.
Lydia Außenhofer 《Mathematische Zeitschrift》2007,257(2):239-250
For a topological group G, we denote by G
a
the arc component of the neutral element and by the character group of G, i.e. the group of all continuous homomorphisms from G into T. We prove the following theorem: Let G be a connected locally compact abelian group and let be the embedding. Then is a topological isomorphism. In particular, the character group of the arc component of a compact abelian group is discrete.
Some conclusions will be drawn. 相似文献
10.
Tsugunori Nogura 《Topology and its Applications》1983,15(1):65-69
We study the question whether a topological space X with a property can be embedded in a countably compact space X? with the same property . 相似文献
11.
Jorge Galindo 《Advances in Mathematics》2004,188(1):51-68
Rosenthal's theorem describing those Banach spaces containing no copy of ?1 is extended to topological groups replacing ?1-basis by interpolation sets in the sense of Hartman and Ryll-Nardzewsky (Colloq. Math. 12 (1964) 23-39). This extension provides a characterization of those locally compact groups containing no interpolation sets and of those locally compact groups which respect compactness, i.e, such that every Bohr compact subset is compact. The approach followed in this paper sheds some light on other questions related to the duality theory of non-Abelian locally compact groups. 相似文献
12.
In this paper we study real lattice homomorphisms on a unital vector lattice
, where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice
of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case
of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces. 相似文献
13.
In this note we prove that every Eberlein compact linearly ordered space is metrizable. (By an Eberlein compact space we mean a topological space which can be embedded as a compact subset of a Banach space with the weak topology.) 相似文献
14.
In this note, we construct an example of a locally compact abelian group
G = C × D (where C is a compact group and D
is a discrete group) and a closed pure subgroup of G
having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question
raised by Hartman and Hulanicki. A simple proof of the following result is given:
Suppose ${\frak K}$ is a class of locally compact abelian groups such
that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and
nG is closed in G for each positive integer
n. If H is a
closed subgroup of a group $G \in {\frak K}$, then
H is topologically pure in
G exactly if the annihilator of
H is topologically pure in
$\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002 相似文献
15.
16.
We study the cardinalities of countably compact, locally countableT
3 spaces. For alln(<), there exists one of cardinality
n
. IfV=L, then there exists one of cardinalityx iffx= orx
=x. MA implies that there exists one of cardinality>2. 相似文献
17.
S. Purisch 《Topology and its Applications》1983,16(3):273-277
Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an Fσ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap. 相似文献
18.
If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M1. 相似文献
19.
Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω1. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω1 is homeomorphic to Iω1. An analogous result is also proved for the cone over Iω1. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved. 相似文献
20.
Let π:X→Y be a surjective continuous map between Tychonoff spaces. The map π induces, by composition, an injective morphism C(Y)→C(X) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C(Y) as a subring of C(X). This paper deals with finiteness properties of the ring extension C(Y)⊆C(X) in relation to topological properties of the map π:X→Y. The main result says that, for X a compact subset of Rn, the extension C(Y)⊆C(X) is integral if and only if X decomposes into a finite union of closed subsets such that π is injective on each one of them. 相似文献