Given a Tychonoff space , a base for an ideal on is called pseudouniform if any sequence of real-valued continuous functions which converges in the topology of uniform convergence on converges uniformly to the same limit. This paper focuses on pseudouniform bases for ideals with particular emphasis on the ideal of compact subsets and the ideal of all countable subsets of the ground space.
Let X, Y be two compacta with Sh(X) = Sh (Y). Then, the spaces of components of X, Y are homeomorphic. This does not happen, in general, when X, Y are quasi-equivalent. In this paper we give a sufficient condition for the existence of a homeomorphism between the spaces of components of two quasi-equivalent compacta X, Y which maps each component in a quasi-equivalent component.
R. Stoltenberg characterized in [2] those quasi-uniformities which are quasi-pseudometrizable, as well as those quasi-metric spaces which have a quasi-metric completion. In this paper we follow Stoltenberg's work by giving characterizations for quasi-metrizability and quasi-metric completion for a particular type of quasi-uniform spaces, the Pervin's quasi-uniform space.
In this article, using mostly Pervin [9], Kunzi [6], [8], [7], Williams [11] and Bourbaki [3] works, we formalize in Mizar [2] the notions of quasiuniform space, semi-uniform space and locally uniform space. We define the topology induced by a quasi-uniform space. Finally we formalize from the sets of the form ((X Ω) × X) ∪ (X × Ω), the Csaszar-Pervin quasi-uniform space induced by a topological space.
Soit un espace topologique régulier et fortement -favorable : si est image continue d’un espace métrisable séparable alors est lusinien; ceci répond à une question de R. Haydon. Si est seulement de Lindelöf et à diagonale alors l’espace mesurable est standard; on en déduit que si l’ensemble des points extrêmaux d’un convexe compact est de Lindelöf et à diagonale , alors est métrisable.