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	~~==Examples==~~

	~~===Line with two origins===~~
	~~The most familiar non-Hausdorff manifold is~~ the ~~'''line with two origins'''~~, ~~or '''bug-eyed line'''~~.

	~~This is~~ the ~~[[quotient space]] of two copies of~~ the ~~real line~~
	:'~~''R''' × {''a''}~~ and ~~'''R''' × {''b''}~~
	~~with~~ the [~~[equivalence relation~~]]
	~~:<math>(x~~,a~~) \sim (x,b)\text{ if }x \neq 0~~.~~\;</math>~~

	This ~~space has a single point for each nonzero real number ''r''~~ and ~~two points 0~~<~~sub~~>~~''a''~~<~~/sub~~> and ~~0<sub>''b''</sub>~~. ~~In this space all neighbourhoods~~ of 0<~~sub~~>~~''a''~~<~~/sub~~> ~~intersect all neighbourhoods~~ of ~~0<sub>''b''</sub>, so it~~ is ~~non~~-~~Hausdorff~~.

	~~Further~~, the ~~line with two origins does~~ not ~~have~~ the ~~homotopy type~~ of a [~~[CW-complex]~~], or of ~~any Hausdorff space~~.<~~ref~~>~~Gabard, pp~~. ~~4–5<~~/~~ref>~~

	=~~==Branching line===~~
	~~Similar~~ to the ~~line~~ with ~~two origins~~ is the ~~'''branching line'''~~.

	This is the ~~[[quotient space]] of two copies of the real line~~
	:'~~''R''' × {''a''}~~ and ~~'''R''' × {''b''}~~
	with ~~the~~ [~~[equivalence relation~~]]
	:<~~math~~>(x,~~a) \sim (x~~,~~b)\text{ if }x~~ < ~~0.\;~~<~~/math~~>

	~~This space has~~ a ~~single point for each negative real number~~ '~~'r'' and two points~~ <~~math~~>~~x_a, x_b~~<~~/math~~> ~~for every non-negative number: it has~~ a ~~"fork" at zero~~.

	~~===Etale space===~~
	The ~~[[etale space]]~~ of a [~~[sheaf (mathematics)\|sheaf]~~], ~~such as the sheaf of continuous real functions over a manifold~~, ~~is a manifold~~ that is ~~often non-Hausdorff~~. ~~(The etale space is Hausdorff~~ if ~~it is a sheaf~~ of ~~functions with some sort~~ of ~~[[analytic continuation]] property~~.~~){{Citation needed\|date=July 2010}}~~

	~~==Notes==~~
	<~~references/~~>

	~~==References==~~
	*{{citation
	~~\|title=Manifolds~~: ~~Hausdorffness versus homogeneity~~
	~~\|first=Mathieu~~
	~~\|last=Baillif~~
	~~\|first2=Alexandre~~
	~~\|last2=Gabard~~
	~~\|arxiv=math~~.GN/~~0609098v1~~
	}}

	*{{citation
	~~\|title~~=~~A separable manifold failing~~ to ~~have the homotopy type of~~ a ~~CW-complex~~
	~~\|first=Alexandre~~
	~~\|last=Gabard~~
	~~\|arxiv=math~~.~~GT/0609665v1~~
	}}

	~~[[Category:Manifolds]]~~
	~~[[Category:Topology]]~~
	~~[[Category:General topology]]~~

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