‎Let ${X_{alpha}:alphainLambda}$ be a collection of topological spaces‎, ‎and $mathcal {G}_{alpha}$ be a grill on $X_{alpha}$ for each $alphainLambda$‎. ‎We consider Tychonoffrq{}s type Theorem for $X=prod_{alphainLambda}X_{alpha}$ via the above grills and a natural grill on $X$ ‎related to these grills, and present a simple proof to this theorem‎. ‎This immediately yields the classical theorem...

In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming 20 = 21 , there exists a normal absolutely star countable discrete clos...

The three-dimensional compressible magnetohydrodynamic isentropic flow with zero magnetic diffusivity is studied in this paper. The vanishing magnetic diffusivity causes significant difficulties due to the loss of dissipation of the magnetic field. The existence and uniqueness of local-in-time strong solutions with large initial data is established. Strong solutions have weaker regularity than ...

We define a topo-system on group as set of subgroups satisfying certain topology-like conditions. study some fundamental properties groups equipped with (topo-groups) and using suitable formalization subgroup filter, we prove Tychonoff type theorem for the direct product topo-compact topo-groups.

In this note the Stone-Cech compactification is used to produce short proofs of two theorems on the structure of free topological groups. The first is: The free topological group on any Tychonoff space X contains, as a closed subspace, a homeomorphic copy of the product space X". This is a generalization of a result of B. V. S. Thomas. The second theorem proved is C. Joiner's, Fundamental Lemma.