Connectedness and path-connectedness
Why this is asked: Connected subsets of ℝ are intervals; continuous images stay connected; path-connected ⇒ connected with the topologist's sine curve as the standard converse failure — except open subsets of ℝⁿ, where the two agree.
Connectedness — tests, preservation and the sine curve
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The trap here
“Connected ⇒ path-connected” — false
Topologist's sine curve {(x, sin 1/x) : 0 < x ≤ 1} ∪ {0}×[−1,1]
Connected as the closure of a connected set; no path reaches the segment.
Check yourself — select all that apply
Which of the following are connected?
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Open this in the full syllabus view · Unit 1