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Every plane domain whose complement contains at least two points is uniformized by the unit disc. A plane universal cover would give a nonconstant entire function omitting two values, contrary to the Little Picard theorem.

The quotients of $\mathbb{C}$ by free properly discontinuous conformal actions are $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, and the complex tori $\mathbb{C}/\Lambda$. The translation group has respectively rank zero, one, or two.

The sphere has the Möbius group $PG{L}_2(\mathbb{C})$, the plane has the affine maps $z\mapsto az+b$ with $a\ne 0$, and the disc has the maps $e^{i\theta }(z-a)/(1-\bar{a}z)$ with $|a|<1$.

Every nonempty simply connected proper open subset of the complex plane is conformally equivalent to the unit disc.

The wired uniform spanning forest of an infinite graph is the weak limit of uniform spanning trees on finite exhaustions whose exterior vertices are identified to one wired boundary vertex.

If $n\ge 2k$ and $\mathcal{A}\subseteq [n{]}^{(k)}$ is intersecting, then $|\mathcal{A}|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$.

A sequence $f_n:X\to Y$ converges locally uniformly to $f$ when every point has a neighbourhood on which the convergence is uniform.

A sequence of functions is uniformly Cauchy when, for every $\epsilon >0$, all sufficiently late pairs satisfy

A uniform limit of uniformly continuous maps between metric spaces is uniformly continuous. Approximate the limit at both endpoints by one fixed member of the sequence and use its uniform continuity between them.

Every continuous map from a compact metric space to a metric space is uniformly continuous.

Independently colour the vertices of a countable graph red or blue with equal probabilities. If every vertex has infinite degree, then each fixed vertex has infinitely many neighbours of each colour almost surely. A countable union of the exceptional null events is null, so such a colouring exists.

Take unfriendly partitions of an exhaustion by finite induced subgraphs and use a diagonal subsequence to stabilize the colour of each vertex. Local finiteness makes every neighbourhood stabilize after finitely many steps, so each unfriendly inequality passes to the limit.

Every finite graph has an unfriendly partition. In a maximum cut, moving any vertex to the other side cannot increase the number of crossing edges, which is exactly the required neighbour inequality.

The Stone-Cech compactification $\beta \mathbb{N}$ can be identified with the space of ultrafilters on $\mathbb{N}$. Its basic clopen sets are $\bar{A}=\{\mathcal{U}:A\in \mathcal{U}\}$.

This is the natural-number-space case of Stone-Čech compactification.

A nonprincipal ultrafilter contains no finite set. On an infinite set it contains the cofinite filter.

The principal ultrafilter at $x\in X$ is $\{A\subseteq X:x\in A\}$. An ultraproduct by a principal ultrafilter is isomorphic to the factor indexed by $x$.

For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matricesThe two squared frequencies and displacement ratios areGravity shifts the equilibrium but does not enter these vertical frequencies.

The semiclassical action integralis strictly increasing for a confining one-dimensional potential. Consequently the two-turning-point WKB quantization condition has at most one energy at each quantum number.

Normal ordering places creation operators to the left of annihilation operators. For free fields it removes vacuum self-contractions from a composite operator.

The holomorphic stress-energy tensor generates infinitesimal holomorphic conformal transformations through its operator product expansions.