The fundamental group of a union of path-connected spaces sharing a basepoint is given by the free product of the fundamental groups of the individual spaces.

Algebraic Topoloy

ker f -> ker g -> ker h -> coker f -> coker g -> coker h is exact

Algebra

For an integrable function, the integral over a closed interval is equal to the difference of the antiderivtive evaluated at the endpoints

Calculus

If every chain has a maximal element, then there exists a maximal element for the poset

Math Foundations

The order of a subgroup divides the order of the group in finite groups

Algebra

Every bounded entire function is constant.

Complex Analysis

The product of any number of compact spaces is compact.

Topology

Every number has a unique decomposition into prime factors

Number Theory

Every nonconstant real polynomial has a complex root.

Algebra/Complex Analysis

Every map D^n->D^n has a fixed point.

Algebraic Topology

A Sphere is equidecomposable with two disjoint spheres.

Set-Theoretic Geometry

There is a bijective correspondence between the subfield of a field extension and teh subgroups of the associated Galois group.

Algebra

Statement

Theorem

Field

Every contraction between metric spaces has a fixed point.

Analysis

If f is an analytic complex function and $/gamma$ is a closed, rectifiable path homologous to 0, then the integral of f over $/gamma$ is 0

Complex Analysis

Nonconstant analytic maps map open sets to open sets.

Complex Analysis

For three compact compact subspaces of R^3, there is a single plane that bisects all three compact subsets.

Algebraic Topology

Every finitely generated abelian group is isomorphic to a unique direct product of cyclic groups.

Algebra

Every finite group has at least one p-group for every prime p dividing the order of the group

Algebra

An infinite collection of first-order sentences has a model iff every finite subset of it has a model

Logic

Every bounded sequence has a convergent subsequence.

Analysis

A subset of R^n is compact iff it is closed and bounded

Analysis

For any group homomorphism f:G->H, G/ker f is isomorphic to im f.

Algebra

For sets A and B, if there are injective maps A->B and B->A, then A and B have the same cardinality.

Set Theory

Every complete metric space is a Baire space, and every locally compact Hausdorff space is a Baire Space

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