In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.
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
\( {\displaystyle (x,a)\sim (x,b){\text{ if }}x\neq 0.} \)
This space has a single point for each nonzero real number r and two points 0a and 0b. A local base of open neighborhoods of \( {\displaystyle 0_{a}} \) in this space can be thought to consist of sets of the form \( {\displaystyle \{r\in \mathbb {R} \setminus \{0\}\vert -\varepsilon <r<\varepsilon \}\cup \{0_{a}\}} \) , where \( \varepsilon \) is any positive real number. A similar description of a local base of open neighborhoods of \( {\displaystyle 0_{b}} \) is possible. Thus, in this space all neighbourhoods of 0a intersect all neighbourhoods of 0b, 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.[1]
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
\( {\displaystyle (x,a)\sim (x,b){\text{ if }}x<0.} \)
This space has a single point for each negative real number r and two points \( x_{a},x_{b} \) for every non-negative number: it has a "fork" at zero.
Etale space
The etale space of a 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.)[2]
Notes
Gabard, pp. 4–5
Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. New York: Springer-Verlag. p. 164. ISBN 978-0-387-90894-6.
References
Baillif, Mathieu; Gabard, Alexandre, Manifolds: Hausdorffness versus homogeneity, arXiv:math.GN/0609098v1
Gabard, Alexandre, A separable manifold failing to have the homotopy type of a CW-complex, arXiv:math.GT/0609665v1
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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