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A simple continued fraction is a continued fraction with all partial numerators equal to $1$ and positive integer partial denominators after the first term. The definition of a simple continued fraction gives the precise form.

A derangement is a permutation that moves every element. Derangements answer the classical hat-check problem, and their number is a standard application of inclusion and exclusion. The node derangement gives the precise definition.

A graph is $k$-colourable when it has a proper colouring with $k$ colours. The least such $k$ is the chromatic number. The node k-colourable graph gives the precise definition.

A rooted tree is a tree with a chosen vertex, the root, which gives each edge a direction away from it. Rooted trees model hierarchies and recursive data. The node rooted tree gives the precise definition.

A cycle in a graph is a closed walk that visits each of its vertices once and returns to its start. Cycles are central in graph theory, for example in the definitions of trees and of bipartite graphs. The node cycle gives the precise definition.

Pell's equation is the Diophantine equation $x^2 - d y^2 = 1$, where $d$ is a positive integer that is not a perfect square. The definition of Pell's equation gives the precise form.

Trial division is the simplest method to factor an integer $n > 1$ or to test whether it is prime. It divides $n$ by the integers $d = 2, 3, 4, \dots$ in turn, and it stops at the first $d$ that divides $n$, or when $d^2 > n$. If no $d$ with $d^2 \le n$ divides $n$, then $n$ is prime.

A bipartite graph is a graph whose vertices split into two parts with every edge between the parts. Bipartite graphs model relations between two kinds of object, and they are the setting of matching theory. The node bipartite graph gives the precise definition.

A topological embedding is a map that identifies a space with a subspace of another space, with the same topology. It is a homeomorphism onto its image.

A map between metric spaces is Hölder continuous with exponent $\alpha$ when $d(f(x), f(y)) \le C \, d(x, y)^\alpha$ for some constant $C$ and all $x, y$. For $\alpha = 1$ this is the Lipschitz condition.

A monotone sequence is a real sequence that never decreases or never increases. A bounded monotone sequence always converges. The node monotone sequence gives the precise definition.

A topological space is regular when each closed set and each point outside it lie in disjoint open sets.

A topological space is second-countable when its topology has a countable basis. This condition limits the size of the topology and holds for Euclidean spaces.

A real sequence assigns a real number to each natural number. Real sequences are the first setting for limits, convergence and series in real analysis. The node real sequence gives the precise definition.

A T0 space, or Kolmogorov space, is a topological space in which the open sets can tell any two distinct points apart. It is the weakest of the usual separation axioms.

A sequence of functions converges uniformly when the largest distance between $f_n$ and the limit tends to zero. Uniform limits keep continuity, and they commute with integration. The node uniform convergence gives the precise definition.

A function has a local minimum at a point when its value there is at most its values at all nearby points. Local minima are the bottoms of the valleys of the graph. The node local minimum gives the precise definition.

A bounded sequence is a real sequence whose terms all lie in one fixed interval. Boundedness is a necessary condition for convergence, and it is the hypothesis of the Bolzano–Weierstrass theorem. The node bounded sequence gives the precise definition.

A strictly convex function is a function whose graph lies strictly below each chord, except at the endpoints of the chord. Its graph contains no line segments. The node strictly convex function gives the precise definition.

A convergent series is a series whose sequence of partial sums has a finite limit. Many tests decide whether a given series converges. The node convergent series gives the precise definition.

A function satisfies a Lipschitz condition when the distance between two values is at most a constant times the distance between the two points. This is a strong form of continuity. In the theory of differential equations, it gives uniqueness of solutions. See the definition.

A sequence of functions is uniformly Cauchy when its terms become uniformly close to each other. For real-valued functions, this is the same as uniform convergence. The node uniformly Cauchy sequence of functions gives the precise definition.

A contractive sequence is a real sequence whose steps shrink at least by a fixed factor less than $1$. Such a sequence is Cauchy, so it converges. The node contractive sequence gives the precise definition.

A topological space is first-countable when each point has a countable family of neighbourhoods that controls all neighbourhoods of the point. In such spaces, sequences are enough to describe closures and continuity.

A geometric sequence is a sequence in which each term is a fixed multiple of the term before it. The fixed factor is the common ratio. The node geometric sequence gives the precise definition.

A series of functions converges normally when the series of the suprema of the absolute values of its terms converges. Normal convergence implies uniform and absolute convergence. The node normal convergence gives the precise definition.

A subsequence of a sequence keeps infinitely many of its terms, in their original order. Subsequences are the main tool of the Bolzano–Weierstrass theorem. The node subsequence gives the precise definition.

A function has a local maximum at a point when its value there is at least as large as its values at all nearby points. Local maxima are the tops of the hills of the graph. The node local maximum gives the precise definition.

A locally compact space is a topological space in which each point has a compact neighbourhood. Such spaces look like compact spaces near each point, as $\mathbb{R}^n$ does.

A telescoping series is a series whose terms have the form $a_n = b_n - b_{n+1}$ for a sequence $(b_n)$. Its partial sums collapse: $\sum_{n=0}^{N} (b_n - b_{n+1}) = b_0 - b_{N+1}$. Thus the series converges if and only if $(b_n)$ converges, and then its sum is $b_0 - \lim b_n$.

A saddle point of a differentiable function of several variables is a critical point that is not a local extremum. Near a saddle point, the function takes values both larger and smaller than its value at the point.

A divergent series is a series whose partial sums do not have a finite limit. Summation methods give values to some divergent series. The node divergent series gives the precise definition.

A differentiable function is a function that has a derivative at each point of its domain. Near each point, it is well approximated by a linear function. The node differentiable function gives the precise definition.

A function is uniformly continuous when one input tolerance $\delta$ controls the output error $\varepsilon$ at all points of the domain at once. This is stronger than continuity at each point. The node uniform continuity gives the precise definition.

A basis of a topology is a family of open sets such that every open set is a union of members of the family. A basis lets you describe a topology by a small set of simple open sets, such as open intervals or open balls.

A series converges conditionally when it converges but the series of absolute values diverges. The sum of such a series depends on the order of its terms. The node conditional convergence gives the precise definition.

A normal space is a topological space in which any two disjoint closed sets can be put into disjoint open sets. In normal spaces, there are many continuous real functions, by Urysohn's lemma.

A T1 space is a topological space in which, for any two distinct points, each one has an open set that misses the other. Equivalently, every one-point set is closed.

A divergent sequence is a sequence that has no limit. Its terms can grow without bound or oscillate. The node divergent sequence gives the precise definition.

A conservative vector field is a vector field that is the gradient of a scalar function, called a scalar potential of the field. Its line integrals depend only on the end points of the path, and they vanish around each closed path.