Related papers: Finite subset spaces of closed surfaces
The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of…
The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of…
Given a topological space X denote by exp_k(X) the space of non-empty subsets of X of size at most k, topologised as a quotient of X^k. This space may be regarded as a union over 0 < l < k+1 of configuration spaces of l distinct unordered…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
This paper expands on and refines some known and less well-known results about the finite subset spaces of a simplicial complex $X$ including their connectivity and their top homology groups. It also discusses the inclusion of the…
A topological space (not necessarily simply connected) is said to have finite homotopy rank-sum if the sum of the ranks of all higher homotopy groups (from the second homotopy group onward) is finite. In this article, we characterize the…
For every finite closure space $X$ one can define a finite topological space $\operatorname{Top} X$ together with a natural projection $\operatorname{Top} X\longrightarrow X$. This could allow to apply the techniques of topological…
We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However,…
Using the topological technique of diagrams of spaces, we calculate the homology of the union and the complement of finite arrangements of subspaces of the form $D + SP^{n-d}(X)$ in symmetric products $SP^n(X)$ where $D\in SP^d(X)$. As an…
We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that…
We show that for an m-connected cell complex X the space exp_k X of non-empty subsets of X of cardinality at most k is (m + k - 2)-connected
Let ${\rm Fin}(X)$ be the hyperspace consisting of non-empty finite subsets of a space $X$ endowed with the Vietoris topology. In this paper, we characterize a metrizable space $X$ whose hyperspace ${\rm Fin}(X)$ is homeomorphic to the…
Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \neq char(\tilde{k}), the cohomology groups H^i_c (\bar{U},…
A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous…
The topology of a space $X$ is generated by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq X$ is closed in $X$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. A space $X$ is a $k$-space…
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…
In this paper we focus on the set-open topologies on the group $\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we…