Abdul Mohamad

On the metrizability of spaces with a sharp base

Abstract A base B for a space X is said to be sharp if, whenever x ∈ X and ( B n ) n ∈ ω is a sequence of pairwise distinct element of B each containing x , the collection {⋂ j⩽n B j : n∈ω} is a base at the point x . We answer questions raised by Alleche et al. and Arhangelskii et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a sharp base and [0,1] need not have a sharp base. We prove various metrization theorems and provide a characterization along the lines of Ponomarevs for point countable bases.

SURFACE DIAGRAMS WITH AT MOST TWO TRIPLE POINTS

In this paper, we prove that if a surface diagram of a surface-knot has at most two triple points and the lower decker set is connected, then the surface-knot group is isomorphic to the infinite cyclic group.

Symmetric g-functions

Abstract A number of generalizations of metrizability have been defined or characterized in terms of g -functions. We study symmetric g -functions which satisfy the condition that x ∈ g ( n , y ) iff y ∈ g ( n , x ). It turns out that the majority of symmetric g -functions fall into one of four known classes of space. Some metrization theorems are proved.

ON TRIPLE POINT NUMBERS OF 5-COLORABLE 2-KNOTS

In this paper we give a lower bound of triple point numbers of special family of 2-knots colored by the dihedral quandle of order 5.

Some Properties of Fuzzy Quasimetric Spaces

Some properties of fuzzy quasimetric spaces are studied. We prove that the topology induced by any 0𝑥0005𝑒𝑀-complete fuzzy-quasi-space is a 0𝑥0005𝑒𝑑-complete quasimetric space. We also prove Baires theorem, uniform limit theorem, and second countability result for fuzzy quasi-metric spaces.

A metrisation theorem for pseudocompact spaces

In this paper we prove that a completely regular pseudocompact space with a quasi‐regular‐G -diagonal is metrizable.

A Result on ℵ1-Compact Spaces

We prove that every countably compact space with quasi-S1-diagonal is compact. However, it is shown that it need not be metrizable.