In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. [1] Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected. [2]
Every ultraconnected space is path-connected (but not necessarily arc connected). If and are two points of and is a point in the intersection , the function defined by if , and if , is a continuous path between and . [2]
Every ultraconnected space is normal, limit point compact, and pseudocompact. [1]
The following are examples of ultraconnected topological spaces.
In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. [1] Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected. [2]
Every ultraconnected space is path-connected (but not necessarily arc connected). If and are two points of and is a point in the intersection , the function defined by if , and if , is a continuous path between and . [2]
Every ultraconnected space is normal, limit point compact, and pseudocompact. [1]
The following are examples of ultraconnected topological spaces.