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Connectedness
Undergraduate
Definition
A topological space is connected if it cannot be separated into two disjoint nonempty open sets.
Formulas
X ≠ U ∪ V for disjoint nonempty open U, V
Definition of connectedness
Examples
Example 1
Explain why ℝ is connected.
Example 2
Show ℚ is not connected.
Applications
Analysis
Intermediate value theorem
Graph Theory
Connected graphs
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