line with two origins



topology (point-set topology, point-free topology)

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What is called the line with two origins is the topological space which results by “gluing” a copy of the real line identically to itself, except at the origin. This is a basic example of a non-Hausdorff topological space (even a non-Hausdorff smooth manifold) and of the fact that quotient topological spaces of Hausdorff topological spaces need not be Hausdorff themselves.


Consider the disjoint union \mathbb{R} \sqcup \mathbb{R} of two copies of the real line \mathbb{R} regarded with its metric topology. Moreover, consider the equivalence relation on the underlying set which identifies every point x ix_i in the iith copy of \mathbb{R} (i{0,1}i \in \{0,1\}) with the corresponding point in the other, the (1i)(1-i)th copy, except when x=0x = 0:

(x iy j)((x=y)and((x0)or(i=j))). \left( x_i \sim y_j \right) \;\Leftrightarrow\; \left( \left( x = y \right) \,\text{and}\, \left( \left( x \neq 0 \right) \,\text{or}\, \left( i = j \right) \right) \right) \,.

The quotient topological space

()/ \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim

by this equivalence relation is called the line with two origins.


The line with two origins is a non-Hausdorff topological space:

Because by definition of the quotient space topology, the open neighbourhoods of 0 i()/0_i \in \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim are precisely those that contain subsets of the form

(ϵ,ϵ) i(ϵ,0){0 i}(0,ϵ). (-\epsilon, \epsilon)_i \;\coloneqq\; (-\epsilon,0) \cup \{0_i\} \cup (0,\epsilon) \,.

But this means that the “two origins” 0 00_0 and 0 10_1 may not be separated by neighbourhoods, since the intersection of (ϵ,ϵ) 0(-\epsilon, \epsilon)_0 with (ϵ,ϵ) 1(-\epsilon, \epsilon)_1 is always non-empty:

(ϵ,ϵ) 0(ϵ,ϵ) 1=(ϵ,0)(0,ϵ). (-\epsilon, \epsilon)_0 \cap (-\epsilon, \epsilon)_1 \;=\; (-\epsilon, 0) \cup (0, \epsilon) \,.

The Hausdorff reflection

H:TopTop Haus H \;\colon\; Top \longrightarrow Top_{Haus}

of the line with two origins is the real line itself:

H(()/) H\left( \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim \right) \;\simeq\; \mathbb{R}

However, it is T1 and sober.

It is also a topological manifold, even a smooth manifold, except that one often requires such to be Hausdorff to rule out examples such as this. In a similar vein, it is locally compact (and a compact version can be made by starting with [1,1][{-1,1}] instead of \mathbb{R}), paracompact, and so forth.


Revised on April 14, 2017 17:46:15 by Toby Bartels (