General Topology by Tom Leinster (2014–15)

By Tom Leinster.

Online text: webhomes.maths.ed.ac.uk/~tl/topology/topology_notes.pdf

Cited passages: Tom Leinster, General Topology, Example C3.6, p. 78, and the discussion after Lemma C4.12, p. 84: the sine-curve obstruction and nonclosed path components; here the positive graph is restricted to $0<x\le 1$; Tom Leinster, General Topology, Lemma C1.8, p. 70: a continuous image is connected; Tom Leinster, General Topology, Lemma C4.12 and the following definition, p. 84: path equivalence and maximal path-connected subsets; Tom Leinster, General Topology, Remark C1.2, p. 68, and Proposition C2.6, p. 73: the nonempty convention and the interval characterization.