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For a nonnegative function $Z$ of independent random variables, entropy is at most the sum of its conditional coordinate entropies. This tensorization is a basic step of the entropy method.

A nonnegative function $f$ of independent coordinates is self-bounding when there are coordinate-deleted versions $f_i$ such that $0\le f-f_i\le 1$ andThese inequalities make the natural variance proxy no larger than the function itself.

An integer-valued function $f$ is $g$-certifiable when every input $x$ with $f(x)=k$ has a set of at most $g(k)$ coordinates whose values alone guarantee that $f\ge k$.

A function $f(x_1,\dots ,x_n)$ has bounded differences with constants $c_i$ when changing only $x_i$ changes its value by at most $c_i$.

For every $\lambda >0$, Markov inequality applied to $e^{\lambda X}$ givesOptimizing over $\lambda$ gives the Chernoff bound in terms of the moment-generating function of $X$.

The entropy of a binary mixture also satisfieswhere $H$ is binary entropy. This follows by applying the operator monotonicity of logarithm to $p{\rho }_1\le p{\rho }_1+(1-p){\rho }_2$ and its counterpart for ${\rho }_2$.

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.

In a language with unary predicates $A,B$, the full class of structures in which exactly one of $A,B$ is infinite is not axiomatizable by a first-order theory. Suppose $T$ axiomatizes that class. Add constants $a_i$ for distinct members of $A$ and constants $b_i$ for distinct members of $B$.

Every finite fragment is satisfiable: take $A$ infinite and $B$ finite but large enough to contain its finitely many required distinct witnesses. This structure belongs to the full class and hence satisfies all of $T$. The compactness theorem then supplies a model of $T$ in which both predicates are infinite, a contradiction.

The same argument applies to definable sets whenever these finite requirements can be realized within the proposed class. It does not forbid axiomatizing a more restricted subclass: for example, the requirement that $A$ be infinite and $B$ empty is axiomatizable by an infinitude schema and one emptiness sentence.

The class of all partially ordered sets with finitely many maximal elements is not axiomatizable by a first-order theory. Suppose $T$ axiomatizes this full class. Add constants $c_0,c_1,\dots$, require them pairwise distinct, and require each to be maximal:Every finite subset is realized in a finite antichain, which has as many maximal elements as needed and is a model of $T$. By the compactness theorem, the enlarged theory has a model with infinitely many distinct maximal elements, contradicting the definition of $T$.

The full-class qualification is essential. The class with at most one maximal element is first-order axiomatizable relative to the partial-order axioms. The obstruction is the unbounded finite number permitted by the original class.

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.