Jest to strona testowa o topologii z zastosowaniem MathJax
A function $f:X\to Y$ is called d-open whenever $f(U)\subseteq\int\cl f(U)$ for any open nonemty $U\subseteq X.$
$\textbf{Theorem.}$
If $f:X\to Y$ is d-open and closed map and $X$ is Tychonoff, then $f$ is open map.
$\textit{proof:}$
Now we shall prove that $f$ is open map. Let $U\subseteq X$ be an open subset and let $x\in U$.
There exists an open neighbourhood $V$ of $x$ such that $x\in V\subset\cl V\subseteq U$. Then
\[f(x)\in f(V)\subseteq \int \cl f(V)=\int f(\cl V)\subseteq f(\cl V)\subseteq f(U),\]
this completes the proof.
$\Box$
A topological space $X$ is $\kappa$-metrizable if there exists a non-negative function
$\rho:X\times\RC(X)\to[0,\infty)$ satisfying the following axioms
(K1) $\rho(x,C)=0$ if and only if $x\in C$ for any $x\in X$ and $C\in \RC(X)$,
(K2) If $C\subseteq D$, then $\rho(x,C)\geq \rho(x,D)$ for any $x\in X$ and $C,D\in\RC(X)$,
(K3) $\rho(\cdot, C)$ is a continuous function for any $x\in X$,
(K4) $\rho(x,\cl(\bigcup_{\alpha<\lambda} C_\alpha))=\inf_{\alpha<\lambda}\rho(x,C_\alpha)$ for any
non-decreasing totally ordered sequence $\{C_\alpha:\alpha<\lambda\}$ and any $x\in X$.