On monotonous separately continuous functions

<p>Let T = (<strong>T</strong>, ≤) and T₁= (<strong>T</strong>₁ , ≤₁) be linearly ordered sets and X be a topological space. The main result of the paper is the following: If function ƒ(t,x) : <strong>T</strong> × X → <strong>T</strong>₁is continuous in each variable (“t” and “x”) separately and function ƒ_x(t) = ƒ(t,x) is monotonous on <strong>T</strong> for every x ∈ X, then ƒ is continuous mapping from<strong> T</strong> × X to <strong>T</strong>₁, where <strong>T</strong> and <strong>T</strong>₁ are considered as topological spaces under the order topology and <strong>T</strong> × X is considered as topological space under the Tychonoff topology on the Cartesian product of topological spaces <strong>T</strong> and X.</p>

On Orthogonally Additive Functions With Big Graphs

Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

Strengthening connected Tychonoff topologies

<p>The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychonoff topology is considered. We show that every Tychonoff space X satisfying ω (X) ≤ c and c (X) ≤ N₀ admits a finer strongly σ-discrete connected Tychonoff topology of weight 2^c. We also prove that every connected Tychonoff space is an open continuous image of a connected strongly σ-discrete submetrizable Tychonoff space. The latter result is applied to represent every connected topological group as a quotient of a connected strongly σ-discrete submetrizable topological group.</p>

A remark on embedding topological groups into products

Let P be a class of topological groups such that every topological group is isomorphic to a topological subgroup of the direct product (with Tychonoff topology) of a subfamily of P. Then every Tychonoff space is homeomorphic to a subspace of a group from P.