Topology counterexamples MOC

Line with two origins

The line with two origins is the real line with two copies of the origin. #m/def/topology Consider the coproduct topology on with inclusions , and let iff . The line with two origins is then

We denote by and by . For all other we have so we may unambiguously refer to .

Properties

• is path-connected but not arc-connected, for any path from to crosses itself.
• is a Non-Hausdorff manifold.

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