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
Topology/Functional Analysis
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