# Bordering of a space

$X$ in a compactification $bX$

A finite family $\{U_1,\ldots,U_k\}$ of sets open in $X$ such that the set $K = X \setminus (U_1 \cup \cdots \cup U_k)$ is compact, and $bX = K \cup \tilde U_1 \cup \cdots \cup \tilde U_k$, where $\tilde U_i$ is the largest open set in $bX$ the intersection of which with $X$ is the set $U_i$ ($X$ is assumed to be completely regular). The concept of a bordering of a space $X$ in $bX$ coincides with the concept of an almost-extendable bordering of a proximity space $X$ (the proximity on $X$ is induced by the extension $bX$), formulated in terms of the proximity: apart from $K$ being compact, it is necessary that for any neighbourhood $O_K$, the family $\{O_k,U_1,\ldots,U_k\}$ is a uniform covering of the space $X$. A bordering of a space $X$ in its Stone–Čech compactification is simply called a bordering of $X$. In the language of borderings, a series of theorems has been formulated on the dimensions of the remainder of compactifications of topological and proximity spaces.

#### References

 [1] Yu.M. Smirnov, "On the dimensions of remainders of compactifications of proximity and topological spaces" Mat. Sb. , 71 : 4 (1966) pp. 554–482 (In Russian)

A concept related to the bordering of a space is that of a border cover: A collection $\mathcal{U}$ of open sets such that $X \setminus \cup \mathcal{U}$ is compact. Border covers work in a sense opposite to borderings. In the case of borderings a compactification is given; from certain systems of border covers one can construct compactifications whose remainders can have special properties.