Connectedness
Objective. Prove connectedness and analyse components.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A topological space X is connected if it cannot be written as the union of two disjoint nonempty open sets, meaning X has no nontrivial 'clopen' (both closed and open) subset besides ∅ and X. A space is path-connected if any two points can be joined by a continuous path, and path-connectedness implies connectedness but not conversely. The connected components of a space are its maximal connected subsets, partitioning X. For example, ℝ∖{0} splits into two connected components, (−∞,0) and (0,∞), because removing the point disconnects the real line into two pieces neither of which can be further split.
Worked example
How many connected components does the subspace S = [0,1] ∪ (2,3) ∪ {5} of ℝ have?
- 1[0,1] is connected as a single interval
- 2(2,3) is connected as a single open interval
- 3{5} is connected trivially as a single point
- 4These three pieces are pairwise separated by gaps, giving 3 connected components
Answer: 3
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Practice
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Start practisingCommon mistake. Students assume connected always implies path-connected, but the topologist's sine curve {(x,sin(1/x)):0<x≤1}∪{(0,y):−1≤y≤1} is connected yet not path-connected, since no continuous path can join the oscillating curve to the segment on the y-axis.
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