References

[1] T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformizability and probabilistic metrizability of probabilistic convergence groups, to appear in Mathematica Slovaca.

[2] T. M. G. Ahsanullah and G. Jäger, Probabilistic convergence transformation groups, submitted for publication, preprint, 2017.

[3] E. Colebunders, H. Boustique, P. Mikusi´nski, G. Richardson, Convergence approach spaces: Actions, Applied Categorical Structures 24(2016), 147–161.

[4] R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37(1997), 257–276.

[5] G. J¨ager, A convergence theory for probabilistic metric spaces, Quaest. Math. 3(2015), 587-599.

[6] G. J¨ager and W. Yao, Quantale-valued gauge spaces, to appear in Iranian Journal of Fuzzy Systems.

[7] H. Lai and W. Tholen, Quantale-valued approach spaces via closure and convergence, arXiv:1604.08813.

[8] R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Clarendon Press, Oxford, 1997. Index Analysis, Springer, 2016.

[9] R. Lowen and B. Windels, Approach groups, Rocky Mountain J. Math. 30(2000), 1057–1073.

]]>For λ = ω it is known that these topologies almost never coincide (*Yamasaki's Theorem).*

In my talk last year, I introduced the *Long Direct Limit Conjecture*, stating that for λ = ω_{1} the two topologies always coincide.

This year, I will introduce one particular example of such a direct limit: The groups of compactly supported homeomorphisms of the Long Line which is naturally such a directed union of topological groups. I will explain why on this group the two direct limit topologies mentioned above agree (and are equal to the compact open topology). Unfortunately this method only works in dimension one and breaks down as soon as one wants to consider groups of homeomorphisms of the Long Plane or similar two dimensional manifolds.

]]>The fiber strong shape theory yields the classification of spaces over B_{0} which is coarser than the classification of spaces over B_{0} induced by fiber homotopy theory, but is finer than the classification of spaces over B_{0} given by usual fiber shape theory.

In the paper [B] an outline of a generalization of Cech homology theory was given by replacing the set of all finite open coverings in the definition of Cech (co)homology group (Ĥ^{n}_{f}(X, A;G)) Ĥ_{n}^{f}(X, A;G) (see [E-S], Ch.IX, p.237) by the set of all finite open families of border open coverings [Sm_{1}].

Following Y. Kodama (see the appendix of [N]), we give the following definition:

**Definition 1**. The border small cohomological dimension d_{∞}^{f}(X;G) of normal space X with respect to group G is defined to be the smallest integer n such that, whenever m ≥ n and A is closed in X, the homomorphism i^{*}_{A, ∞}:Ĥ_{∞}^{m}(X;G)→ [^(H)]_{∞}^{m}(A;G) induced by the inclusion i:A→ X is an epimorphism.

The border small cohomological dimension of X with coefficient group G is a function d^{f}_{∞}:*N*→ N∪{0, + ∞}:X→ n, where d_{∞}^{f}(X;G)=n and N is the set of all positive integers.

We have the following results:

**Theorem 2**. Let X be a metrizable space. Then the following equality d^{f}_{∞}(X;G) = d_{f}(βX\X;G)

holds, where d_{f}(βX\X;G) is the small cohomological dimension of βX\X (see [N], p.199).

**Theorem 3**. Let A be a closed subspace of a normal space X. Then d^{f}_{∞}(A;G) ≤ d^{f}_{∞}(X;G).

**Corollary 4**. For each closed subspace A of a metrizable space X, d^{f}_{∞}(A;G) ≤ d_{f}(βX\X;G).

**Definition 5**. The border large cohomological dimension D_{∞}^{f}(X;G) of normal space X with respect to group G is defined to be the largest integer n such that Ĥ_{∞}^{n}(X, A;G) ≠ 0 for some closed set A of X.

The border large cohomological dimension of X with coefficient group G is a function D_{∞}^{f}:*N*→ N∪{0, + ∞}:X→ n, where D_{∞}^{f}(X;G)=n and N is the set of all positive integers.

**Theorem 6**. For each metrizable space X, one has D_{∞}^{f}(X;G) = D_{f}(βX\X;G),

where D_{f}(βX\X;G) is the large cohomological dimension of βX\X (see [N], p.199).

**Theorem 7**. If A is a closed subset of normal space X, then D_{∞}^{f}(A;G) ≤ D_{∞}^{f}(X;G).

**Corollary 8**. For each closed subspace A of metrizable space X, one has D^{f}_{∞}(A;G) ≤ D_{f}(βX\X;G).

**Theorem 9**. If X is a normal space, then d_{∞}^{f}(X;G) ≤ D_{∞}^{f}(X;G).

**Corollary 10**. For each metrizable space X, one has d_{f}(βX\X;G) ≤ D^{f}_{∞}(X;G)

and d^{f}_{∞}(X;G) ≤ D_{f}(βX\X;G).

**Remark 11**.The results of this paper also hold for spaces satisfying the compact axiom of countability [Sm_{1}]. The locally metrizable spaces, complete in the seance of Cech spaces and locally compact spaces satisfy the compact axiom of countability.

The principal categories of interest to topologists are all of this type, or may be reflectively or coreflectively embedded into them. But as indicated above, an individual category, like Top, may be presentable in various (T, V )-guises, and establishing the equivalence may not necessarily be easy. In fact, its validity may depend on additional properties of V . For example, for T the powerset monad, we may easily extend the usual properties of distance and closure to define and study so-called V -topological spaces, but the establishment of their equivalent description in terms of a V -valued ultrafilter convergence relation requires V to be completely distributive (see Lai and Tholen). Among other theorems, we will present this equivalence statement and show how it unifies previous results for topological spaces and approach spaces and leads to novel applications. Time permitting we will also discuss essential topological properties, like compactness and separation, in the V -context.

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