In this brief note, we prove that every space is a sequence-covering image of a topological sum of convergent sequences. As the application of this result, sequence-covering images of locally compact metric spaces (or, locally separable metric spaces, metric spaces) and sequentially-quotient images of a locally compact metric spaces (or, locally separable metric spaces, metric spaces) are equiv...

In this paper A, B, X, Y denote sets. Let X be an empty set. Note that ⋃ X is empty. Next we state several propositions: (1) (δX) ◦A ⊆ [:A, A :]. (2) (δX) ([:A, A :]) ⊆ A. (3) For every subset A of X holds (δX) ([:A, A :]) = A. (4) dom〈π2(X×Y ), π1(X×Y )〉 = [:X, Y :] and rng〈π2(X×Y ), π1(X×Y )〉 = [:Y, X :]. (5) 〈π2(X × Y ), π1(X × Y )〉 ◦[:A, B :] ⊆ [:B, A :]. (6) For every subset A of X and for...

It is well-known that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compact-open topology. It is less well-known that among arbitrary topological spaces, the exponentiable spaces are precisely the core-compact spaces. The available approaches to the general characterization are based on either category theory or co...

We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximated by finite dimensional quantum compact metric spaces, and that they form a continuous family of quantum compact metric spaces over the space of multipliers of the solenoid, properly metrized.

We present some lifting theorems for continuous order-preserving functions on locally and σ-compact Hausdorff preordered topological spaces. In particular, we show that a preorder space has multi-utility representation if, only every compact subspace, function can be lifted to the entire space. Such characterization is also presented by introducing property of ≾-C-compatible closed subspaces. T...

This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting p...