Fundamental Concepts

Abstract

Let X be a given set. A topology on X is a set $\mathfrak{U}$ of subsets of X satisfying the following properties: A1 $\varnothing, X \in \mathfrak{U}$; A2 if {U _α | α ∈ J} is a set of elements of $\mathfrak{U}$, then

$${\bigcup \nolimits }_{\alpha \in J}{U}_{\alpha } \in \mathfrak{U};$$

A3 if $\{{U}_{\alpha }\vert \alpha = 1,\ldots, n\}$ is a finite set of elements of $\mathfrak{U}$, then

$${\bigcap \nolimits }_{\alpha =1}^{n}{U}_{ \alpha } \in \mathfrak{U}.$$