Topological properties of spaces with a distance

A set $X$ with a distance function $\rho\colon X^2\to\mathbb R_+$ satisfying the identity axiom, that is, such that $\rho(x,y)=0$ if and only if $x=y$, is considered. The function $\rho$ determines a topology on $X$; a set $U\subset X$ belongs to this topology if and only if, for each $u\in U$, there exists a positive $\delta$ such that $\{x\colon \rho(u,x)<\delta\}\subset U$. Closedness, sequential closedness, compactness, sequential compactness, and total boundedness in topological spaces thus arising are studied.