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1.
It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:
- (1)
- the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;
- (2)
- the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.
2.
Julio Becerra Guerrero 《Journal of Functional Analysis》2008,254(8):2294-2302
We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:
- (1)
- Every member of R has the Daugavet property.
- (2)
- It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product lie in R.
- (3)
- If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C(K,(Y,τ)) (of all Y-valued τ-continuous functions on K) is a member of R.
- (4)
- If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C(K,Y)-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R.
- (5)
- All dual Banach spaces without minimal M-summands are members of R.
3.
Jochen Wengenroth 《Journal of Functional Analysis》2003,201(2):561-571
In 1971 Palamodov proved that in the category of locally convex spaces the derived functors Extk(E,X) of Hom(E,·) all vanish if E is a (DF)-space, X is a Fréchet space, and one of them is nuclear. He conjectured a “dual result”, namely that Extk(E,X)=0 for all if E is a metrizable locally convex space, X is a complete (DF)-space, and one of them is nuclear. Assuming the continuum hypothesis we give a complete answer to this conjecture: If X is an infinite-dimensional nuclear (DF)-space, then
- (1)
- There is a normed space E such that Ext1(E,X)≠0.
- (2)
- where is a countable product of lines.
- (3)
- Extk(E,X)=0 for all k?3 and all locally convex spaces E.
4.
We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g.,
- (1)
- if G is a Hausdorff abelian topological group and μ:N2→G is countably additive, then the range μ(N2)={μ(A):A⊆N} is compact metrizable;
- (2)
- if X is a Hausdorff locally convex space and {xj}⊂X, then F={∑j∈Δxj:Δ⊂N, Δ is finite} is relatively compact in (X,weak) if and only if F is relatively compact in X, and if and only if F is relatively compact in (X,F(M)) where F(M) is the Dierolf topology which is the strongest 〈X,X′〉-polar topology having the same subseries convergent series as the weak topology.
5.
For a space X, X2 denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X2. The following are known:
- •
- ω2 is not normal, where ω denotes the discrete space of countably infinite cardinality.
- •
- For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.
- (1)
- ω2 is strongly zero-dimensional.
- (2)
- K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.
6.
Masami Sakai 《Topology and its Applications》2012,159(1):308-314
Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.
Theorem.
For a space X, the following are equivalent:
- (1)
- F[X]is a k-space;
- (2)
- F[X]is sequential;
- (3)
- F[X]is Fréchet-Urysohn;
- (4)
- Every finite power of X is Fréchet-Urysohn for finite sets;
- (5)
- Every finite power ofF[X]is Fréchet-Urysohn for finite sets.
7.
Alessandro Berarducci Dikran Dikranjan Jan Pelant 《Topology and its Applications》2009,156(7):1422-1437
A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X×Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds:
- (a)
- both X and Y are precompact;
- (b)
- both X and Y are locally connected;
- (c)
- one of the spaces is both precompact and locally connected.
8.
Gabriel Padilla 《Topology and its Applications》2007,154(15):2764-2770
A classical result says that a free action of the circle S1 on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H2(B,Z), the Euler class. When the action is not free we have a difficult open question:
- (Π)
- “Is the space X determined by the orbit space B and the Euler class?”
- •
- the intersection cohomology of X,
- •
- the real homotopy type of X.
9.
Vera Toni? 《Topology and its Applications》2010,157(3):674-691
We prove a generalization of the Edwards-Walsh Resolution Theorem:
Theorem.
Let G be an abelian group withPG=P, where. Letn∈Nand let K be a connected CW-complex withπn(K)≅G,πk(K)≅0for0?k<n. Then for every compact metrizable space X with XτK (i.e., with K an absolute extensor for X), there exists a compact metrizable space Z and a surjective mapπ:Z→Xsuch that
- (a)
- π is cell-like,
- (b)
- dimZ?n, and
- (c)
- ZτK.
10.
Mohamed Aziz Taoudi 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):478-3452
In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:
- (i)
- AM is relatively weakly compact;
- (ii)
- B is a strict contraction;
- (iii)
- .
11.
Yankui Song 《Topology and its Applications》2012,159(3):814-817
In this paper, we show the following statements:
- (1)
- For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.
- (2)
- Assuming ℵ02=ℵ12, there exists a centered-Lindelöf normal space X such that we(X)?ω1.
12.
Karim Belaid 《Topology and its Applications》2011,158(15):1969-1975
In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:
- (i)
- D is co-finite in K(X);
- (ii)
- for each x∈K(X)?D, {x} is closed.
13.
Christopher Mouron 《Topology and its Applications》2009,156(3):558-576
Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:
- (1)
- X is non-Suslinean.
- (2)
- If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.
- (3)
- If X is G-like, then X is indecomposable.
- (4)
- If all lie in the same ray and X is finitely cyclic, then X is indecomposable.
14.
Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples:
- -
- a KC space which cannot be embedded in any compact KC space;
- -
- a countable KC space which does not admit any coarser compact KC topology;
- -
- a minimal Hausdorff space which is not a k-space.
15.
The two main results are:
- A.
- If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X∗ is non-separable (and hence X does not embed into c0).
- B.
- There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.
16.
Fréchet-Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative; for instance, the square of a compact F-U space is not in general Fréchet-Urysohn [P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980) 749-753. [27]]. Van Douwen proved that the product of a metrizable space by a Fréchet-Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following:
- (1)
- If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact.
- (2)
- Leaning on (1) we point out a big class of hemicompact sequential non-Fréchet-Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex.
- (3)
- Similar results are also obtained in the framework of locally convex spaces.
17.
Let G be a Hausdorff topological group. It is shown that there is a class C of subspaces of G, containing all (but not only) precompact subsets of G, for which the following result holds:Suppose that for every real-valued discontinuous function on G there is a set A∈C such that the restriction mapping f|A has no continuous extension to G; then the following are equivalent:
- (i)
- the left and right uniform structures of G are equivalent,
- (ii)
- every left uniformly continuous bounded real-valued function on G is right uniformly continuous,
- (iii)
- for every countable set A⊂G and every neighborhood V of the unit e of G, there is a neighborhood U of e in G such that AU⊂VA.
18.
The following results are obtained.
- -
- An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.
- -
- Every hereditarily thickly covered space is aD and linearly D.
- -
- Every t-metrizable space is a D-space.
- -
- X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.
19.
Luoshan Xu 《Topology and its Applications》2006,153(11):1886-1894
In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:
- (1)
- A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;
- (2)
- A poset is continuous iff its Scott topology is completely distributive;
- (3)
- A topological T0 space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;
- (4)
- A topological T1 space is a discrete space iff its topology is completely distributive.
20.
We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:
- (i)
- Characterize those base spaces X and Y for which all isometries are weighted composition maps.
- (ii)
- Give a condition independent of base spaces under which all isometries are weighted composition maps.
- (iii)
- Provide the general form of an isometry, both when it is a weighted composition map and when it is not.