[[Topology MOC]]
# Topology counterexamples MOC
 
The following is a list of rather _pathological_ [[Topological space|topological spaces]] that serve as nice counterexamples.

## Pathological topologies on arbitrary sets

- [[Trivial topology]] is not [[Hausdorff space|Hausdorff]]
- [[Discrete topology]] is [[Completely disconnected]]

## Pathological manifolds

- [[Line with two origins]] is [[Path connectedness|path-connected]] but not [[Path connectedness|arc-connected]] (and not [[Hausdorff space|Hausdorff]])
- [[Long line]]


## Miscellaneous

- [[Topologist's sine curve]] is [[Connectedness|connected]] but not [[Path connectedness|path-connected]]

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