I really liked the Facebook page "Mathematical Theorems you didn't know existed because they're false". I feel like it is a good way to convince oneself of the interest of a counterexample to stare at a theorem that seems quite true and then try to disprove it. I figured I would put here a list of such untrue statements I've encountered (and often believed for a while). Of course, they reflect my interest in geometry, topology and group theory. Most counterexamples to the theorems are constructive in nature and the reader is encouraged to look for them themselves. However, the symbol * indicates counterexamples that are especially hard to find or use existing litterature. These are rather meant for the reader's general interest.
In a CAT(0) metric space, every quasi-geodesic (image of the real line by a quasi-isometric embedding) stays at bounded distance from a geodesic.
A countable disjoint union of closed path-connected non-empty subsets of the Euclidean plane is disconnected.
*Let A be a closed subset of Euclidean 3-space. Assume:
A partition of Euclidean 3-space in circles contains some circle of null radius.
A connected non-empty subset of the Euclidean plane whose path-components are singletons is a singleton.
Let B be a bounded subset in a CAT(0) metric space X. A family of convex subspaces that all intersect B and pairwise intersect in X intersect globally.
Let A be a subset of Euclidean 3-space homeomorphic to the Cantor set. The complement of A is simply connected.
The order of a non-trivial automorphism of the free group with two generators is even or infinite.
A connected, second countable space locally isomorphic to a line is orientable.
Let X be a locally compact Hausdorff space and f : X->X a homeomorphism. Assume that every point x in X has a neighborhood V such that every translate fn(V) is disjoint from V (the action is wandering and free). Then every compact subset K meets finitely many of its translates fn(K) (the action is properly discontinuous).
*Let G be a non-trivial finitely generated group. There is no isomorphism between G and its direct square G2.
An infinite direct product of finite groups cannot act properly on a CAT(0) cube complex.
The unit cube is of least surface area among polyhedra with 8 vertices and volume 1.
A map of Euclidean triangles that is 1-Lipschitz on each edge is 1-Lipschitz on the whole triangle.
A bouquet of two contractible spaces is contractible.
In a closed surface, consider a collection of disjoint simple closed curves that is globally non-separating (cutting all the curves does not disconnect the surface). If the collection is maximal for the inclusion, it has maximal size.