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The complex unit circle is the subgroup $\{z\in \mathbb{C}:|z|=1\}$ under multiplication.

A model is aleph-zero-homogeneous when every finite partial elementary map extends by one more element.

With its standard topology, this type space is a Stone space. The type space $S_n^M(A)$ is the set of complete $n$-types over a parameter set $A$ that are consistent with the complete theory of $M$ together with its diagram over $A$.

A closed subspace of a complete metric space is complete. A Cauchy sequence in the subspace converges in the ambient space, and closedness keeps its limit in the subspace.

The space $C^{\infty }([0,1])$ becomes a complete metric space underConvergence in this metric is exactly uniform convergence of every derivative separately.

Every complete metric space is a Baire space: every countable intersection of open dense subsets is dense. Equivalently, no nonempty open subset is a countable union of nowhere-dense subsets.

To prove the dense-intersection form, let $V$ be nonempty and open and let $G_1,G_2,\dots$ be open dense sets. Successively choose closed ballsstarting with a ball contained in $V\cap G_1$. The centres form a Cauchy sequence. Its limit belongs to every closed ball, and hence to $V\cap {\bigcap }_n{G}_n$.

A very ample divisor is a divisor whose associated line bundle is a very ample line bundle. A divisor is very ample when its complete linear system defines a closed embedding into projective space.

For the complementary modulus $k^{\prime }=\sqrt{1-k^2}$,

Two forcing conditions are incompatible when they have no common stronger extension.

Suppose a class requires exactly one of two definable sets to be infinite. Add constants witnessing arbitrarily many distinct members of both sets. Every finite fragment can keep one set infinite and make the other large but finite, whereas compactness makes both sets infinite. Therefore such an exclusive disjunction is not first-order axiomatizable.

If a theory claimed that a poset has only finitely many maximal elements, adjoining countably many constants required to be distinct and maximal would remain finitely satisfiable. Compactness would produce a model with infinitely many maximal elements, contradicting the claimed axiomatization.

If a theory were inconsistent, a proof of contradiction would use a finite inconsistent subset. Therefore consistency of every finite subset implies consistency of the whole theory, and completeness supplies a model.

A continuous map is proper when inverse images of compact sets are compact. Proper maps induce contravariant maps on compactly supported cohomology.

The imaginary Airy operator $-d^2/d{x}^2+ix$ on $L^2(\mathbb{R})$ has compact resolvent but empty spectrum. Real translations shift it by a purely imaginary scalar, forcing vertical translation invariance of its spectrum.

In a finite chain, a communicating class that is not closed is transient: after it is left, it need not be visited again and is eventually left almost surely.

A communicating class is closed when no transition from it can reach a state outside it.

Two states of a Markov chain communicate when each can be reached from the other with positive probability in finitely many steps.

The necessity direction in Rado's theorem colors an integer using initial data from a P-adic valuation. Applying partition regularity and grouping the coordinates of a monochromatic solution by valuation yields the blocks in the columns property; reduction modulo successively higher powers of $p$ supplies the required linear dependences.

Rado's theorem states that a rational matrix is partition regular if and only if it has the columns property.

A string operator creates excitations at its endpoints. Multiplying by a successive segment moves an endpoint because the shared intermediate excitation is toggled twice.