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Abstract
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Let $\kappa$ be an infinite cardinal,
Let $X$ and $Y$ topological spaces. A function $\phi :X\longrightarrow Y$ is said to be $\kappa$-continuous if for every subspace $A$ of $X$ such that $|A| \leq \kappa$ , the restriction $\phi|_A$ is continuous.
Also a function $\phi :X\longrightarrow Y$ is said to be strictly $\kappa$-continuous if for every subspace $A$ of $X$ such that $|A| \leq \kappa$ , the restriction $\phi|_A$ coincides with the restriction to $A$ of some continuous function $g:X\rightarrow Y$.
$\kappa$-continuous if for every subspace $A$ of $X$ such that $|A| \leq \kappa$ , the restriction $\phi|_A$ coincides with the restriction to $A$ of some continuous function $g:X\rightarrow Y$.
Let $\kappa$ be an infinite cardinal, $X$ be a topological space,
%$\kappa$ is an infinite cardinal and
the \textit{functional tightness of a space $X$} is $t_0(X) = \min\{\kappa :$ every ~ $\kappa$-continuous real-valued function on $X$ is continuous$\}$.\\
The \textit{minitightness} (or the weak functional tightness) of a space $X$ is $t_m(X) = \min\{\kappa : $ every~ strictly~ $\kappa$-continuous real-valued function on $X$ is continuous$\}$.
In this thesis we define, $\Bbb R$-quotient mappings, $\kappa$-close, $\kappa$-dense, \ldots. We study the properties of functional tightness especially in the Tikhonov spaces and find a way to answer this Oleg Okunev question that \lq\lq Is the main functional tightness $X$ equal to functional tightness $X^{\omega}$ for a compact space $X$?''
is the main goal of this thesis.
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