Topology

For any convex polyhedron, the number of vertices $V$, edges $E$, and faces $F$ are related by the equation $V - E + F = 2$.

For any continuous function $f$ mapping a compact convex set to itself in a Euclidean space, there is a point $x_0$ such that $f(x_0) = x_0$.

The product of any collection of compact topological spaces is compact with respect to the product topology.

In a complete metric space (or a locally compact Hausdorff space), the intersection of any countable collection of dense open sets is dense.

In a normal topological space, for any two disjoint closed sets $A$ and $B$, there exists a continuous function $f$ mapping the space to $[0, 1]$ such that $f(A) = \\{0\\}$ and $f(B) = \\{1\\}$.

Every simple closed curve in the plane divides the plane into exactly two connected components: an 'inside' and an 'outside'.

Any continuous function from an $n$-sphere into Euclidean $n$-space maps some pair of antipodal points to the same point.

Any continuous real-valued function defined on a closed subset of a normal topological space can be extended to a continuous function on the whole space.

There is no non-vanishing continuous tangent vector field on an even-dimensional sphere.

A formula that counts the number of fixed points of a continuous mapping from a compact manifold to itself by using traces of the induced mappings on homology groups.

Relates the homology of a subset of a sphere to the cohomology of its complement.

Expresses the fundamental group of a union of two path-connected spaces in terms of the fundamental groups of the spaces and their intersection.

If $U$ is an open subset of $\mathbb{R}^n$ and $f: U \to \mathbb{R}^n$ is an injective continuous map, then $V=f(U)$ is open and $f$ is a homeomorphism between $U$ and $V$.

A map between CW complexes that induces isomorphisms on all homotopy groups is a homotopy equivalence.

A space is compact if and only if every cover by elements of a subbase has a finite subcover.

A strengthening of the Jordan Curve Theorem for the plane, stating that the regions are homeomorphic to the interior and exterior of a unit disk.

Generalization of Brouwer's Fixed-Point Theorem.

Fundamental result in differential topology for manifolds of dimension at least 5.

Refers to homology of quotient spaces (unrelated to the physics one).

A space is metrizable if and only if it is regular and has a $\sigma$-locally finite basis.

The $n$-sphere is simply connected for $n \ge 2$.

Relates homotopy groups and homology groups.

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere (Proven by Perelman).

Geometry of a hyperbolic manifold is determined by its fundamental group.

Relates infinite symmetric products to homology.

Behavior of homotopy groups under suspension.

For $n \ge 5$ (Smale), $n=4$ (Freedman).

Invariant of manifolds (Statement).