Ultraconnected space (original) (raw)

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Property of 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 X {\displaystyle X} {\displaystyle X} is path-connected (but not necessarily arc connected). If a {\displaystyle a} {\displaystyle a} and b {\displaystyle b} {\displaystyle b} are two points of X {\displaystyle X} {\displaystyle X} and p {\displaystyle p} {\displaystyle p} is a point in the intersection cl ⁡ { a } ∩ cl ⁡ { b } {\displaystyle \operatorname {cl} \{a\}\cap \operatorname {cl} \{b\}} {\displaystyle \operatorname {cl} \{a\}\cap \operatorname {cl} \{b\}}, the function f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} {\displaystyle f:[0,1]\to X} defined by f ( t ) = a {\displaystyle f(t)=a} {\displaystyle f(t)=a} if 0 ≤ t < 1 / 2 {\displaystyle 0\leq t<1/2} {\displaystyle 0\leq t<1/2}, f ( 1 / 2 ) = p {\displaystyle f(1/2)=p} {\displaystyle f(1/2)=p} and f ( t ) = b {\displaystyle f(t)=b} {\displaystyle f(t)=b} if 1 / 2 < t ≤ 1 {\displaystyle 1/2<t\leq 1} {\displaystyle 1/2<t\leq 1}, is a continuous path between a {\displaystyle a} {\displaystyle a} and b {\displaystyle b} {\displaystyle b}.[2]

Every ultraconnected space is normal, limit point compact, and pseudocompact.[1]

The following are examples of ultraconnected topological spaces.

  1. ^ a b PlanetMath
  2. ^ a b Steen & Seebach, Sect. 4, pp. 29-30
  3. ^ Steen & Seebach, example #50, p. 74