The theorem gives an exact test for when a topology comes from a distance. Countably many basic open sets are no longer needed. It is enough that the basic open sets come in countably many layers, and that near each point only finitely many sets of each layer are present.

In an ordinary polynomial ring the variable $x$ commutes with every coefficient. In an Ore extension, moving $x$ past a coefficient $a$ changes $a$ to $\sigma(a)$ and adds a correction term $\delta(a)$. Every element can still be written uniquely as $a_0 + a_1 x + \cdots + a_n x^n$, so the ring looks like a polynomial ring, only with a new rule of multiplication. For example, with $R = k[y]$, $\sigma$ the identity and $\delta = d/dy$, the rule $xy = yx + 1$ gives the first Weyl algebra.

Points are concyclic when they all lie on one circle. A polygon with concyclic vertices is a cyclic polygon, and the circle is its circumcircle.

The ordinary fiber of $f$ over $y_0$ collects the points that land exactly on $y_0$. The homotopy fiber is more forgiving: a point $x$ counts if it lands anywhere, as long as one also says how to walk from $f(x)$ to $y_0$. So its points are pairs, a point $x$ and a path from $f(x)$ to $y_0$. If $X$ is a single point mapped to $y_0$, the pairs are just the loops at $y_0$.

Plot the heights of children against their ages, and the points lie roughly along a line. Linear regression describes the height as an intercept plus a slope times the age, plus a random deviation for each child. The slope says how much height increases per year on average. With several explanatory variables, such as age and parents' heights, the line becomes a plane or a higher-dimensional flat surface.

The absolute value of a number is its distance from $0$. We write $|x|$ for the absolute value of $x$. For a real number, $|x|$ is $x$ without its sign, for example $|-3| = |3| = 3$. For a complex number $a + b i$, the absolute value is the modulus $\sqrt{a^2 + b^2}$.

Surreal numbers start from nothing and grow by one rule: a new number sits between a set of numbers on its left and a set on its right. From the empty pair one gets $0$, then $1$ and $-1$, then $2$, $\tfrac12$ and so on. Every real number appears, and so do infinite numbers like $\omega$ and tiny numbers like $1/\omega$. You can still add, subtract, multiply and divide them, and compare any two of them. So they are one huge number line that holds both the reals and the ordinals.

A submartingale tends to go up, or at least not down, on average. Given everything known today, the best forecast of tomorrow's value is at least today's value. The value can still go down on some days; only the average trend is upward. If $S_n$ is a fair coin-tossing walk, then $S_n^2$ is a submartingale: squaring adds $1$ on average at each step.

The field of fractions of an integral domain is the smallest field that contains it. Its elements are the fractions $a/b$ with $b \neq 0$. The rational numbers are the field of fractions of the integers. The definition of the field of fractions gives the construction.

The weight $1/\abs{x}^p$ is large near the origin, so the left side punishes functions that are large near $0$. The inequality says that the gradient energy alone pays for this, as long as the dimension is larger than $p$. In space with $p = 2$, it says that $\int u^2 / \abs{x}^2 \le 4 \int \abs{\nabla u}^2$.

In Łukasiewicz logic a statement can be partly true, with a truth value anywhere between $0$ (false) and $1$ (true). "Not $x$" has truth value $1 - x$, and a kind of "or" adds the truth values but stops at $1$. For example, $0.3 \oplus 0.5 = 0.8$, while $0.7 \oplus 0.5 = 1$. An MV-algebra is any structure that obeys the same equations as these operations on $[0, 1]$.

Each finite Boolean algebra is the algebra of all subsets of some finite set. Each element is the set of atoms below it, and the atoms are the smallest nonzero elements.

Picture a measure as paint spread over a space. Restricting it to a region $A$ wipes away all the paint outside $A$. A set $B$ then gets the amount of paint that lies in both $B$ and $A$. For example, restrict length on the real line to $A = [0, 1]$. Then the interval $[0, 5]$ gets the value $1$, and the interval $[2, 3]$ gets the value $0$.

A variety of general type is, in a precise sense, a variety with a lot of negative curvature. The conjecture predicts that holomorphic copies of the complex plane in such a variety cannot spread everywhere: they all lie inside one smaller algebraic subvariety. In other words, such a variety is Brody hyperbolic outside a small exceptional set.

The coproduct of an element is a sum of pairs, but the pairs are rarely worth naming one by one. Sweedler notation writes a typical pair as $c_{(1)} \otimes c_{(2)}$ and lets the summation stay implicit. With it, the counit law reads $\varepsilon(c_{(1)}) c_{(2)} = c$, and the antipode law reads $S(c_{(1)}) c_{(2)} = \varepsilon(c) 1$. The symbols $c_{(1)}$ and $c_{(2)}$ are not elements on their own; they only make sense inside expressions that are linear in each of them.

Multiplication repeats addition, and powers repeat multiplication. Tetration repeats powers: ${}^{3}2$ means $2^{2^2} = 16$. The values explode: ${}^{4}2 = 65536$, and ${}^{5}2$ is a number with almost twenty thousand digits. Each step applies the map $x \mapsto 2^x$ once more, so tetration is a picture of iterating a function.

A ternary operation takes three elements and gives one. On the integers, $(x, y, z) \mapsto x - y + z$ is an example. On a totally ordered set, the median of three elements, the one in the middle, is another example. The rule "if $x$ then $y$ else $z$" on truth values is a third example.

Shifting a signal later in time by $\lambda$ is, in the field of operators, multiplication by one fixed operator. That operator behaves like an exponential: two shifts compose by adding their lengths. This is why it is written $e^{-\lambda s}$.

A graded module splits into pieces of degree $0, 1, 2, \dots$, and each piece is a finite-dimensional vector space. The theorem says that the dimensions of these pieces are eventually given by a polynomial in the degree. So an infinite list of numbers is controlled by finitely many coefficients.

How can one prove that a polynomial is positive on a region defined by polynomial inequalities? The Positivstellensatz says that there is always an algebraic proof: an identity built from squares and from the defining inequalities. Such an identity can be checked by expanding, without solving anything.