Die Teilmengen X1, X2, X2, …des topologischen Raumes E seien...

Die Teilmengen $X_{1}, X_{2}, X_{2}, \ldots$ des topologischen Raumes $E$ seien zusammenhängend, und die Durchschnitte $X_{n} \cap X_{n+1}(n=1,2, \ldots)$ seien alle nichtleer. Dann ist die Vereinigung $X=\bigcup_{n-1}^{1} X_{n}$ ebenfalls zusammenhängend.

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Key Concepts

Connectedness

Connectedness is a central concept in topology that describes a space which cannot be partitioned into two or more disjoint nonempty open sets. It implies that the space is 'all in one piece', with no separation into isolated parts.

Union of Connected Sets

While the union of connected sets is not necessarily connected, a common sufficient condition is that each set in the collection overlaps with at least one other set in the chain. When connected sets are 'chained together' by nonempty intersections, their union is guaranteed to be connected.

Topological Space

A topological space is a set together with a collection of open sets that satisfy certain axioms (such as closure under arbitrary unions and finite intersections). This framework is fundamental for discussing properties like continuity, convergence, and various forms of connectivity within the space.

Intersection Property

The intersection property refers to the requirement that consecutive sets in a sequence have nonempty intersections. This condition is crucial in ensuring that there is no gap between the connected subsets, thereby allowing their union to inherit the connectedness property.

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