In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two...

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may...

open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of...

{\displaystyle S} of a topological space X {\displaystyle X} is called a regular open set if it is equal to the interior of its closure; expressed symbolically,...

mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations...

mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its...

a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is...

dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open. The empty...

closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where...

cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set. Covers are...