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A continuous map $F:X\to X$ is topologically mixing if for every pair of nonempty open sets $U,V$ there is an integer $N$ such that $F^n(U)\cap V\ne \varnothing$ for every $n\ge N$. This is stronger than topological transitivity, which asks only for some hitting time. For the full tent map, each sufficiently small dyadic branch interval is mapped onto the whole space by an iterate, proving mixing.

A topological space is first-countable when each point has a countable neighbourhood basis. Every metric space is first-countable: the open balls $B(x,1/n)$ form such a basis. A subset of a first-countable space remains first-countable in its subspace topology, since intersecting a local basis with the subset preserves the defining containment property. Thus a space failing first countability cannot be homeomorphic to any subset of a metric space.

Choose an uncountable set $X$ and a point $a\in X$. Every point other than $a$ is isolated, and a set containing $a$ is open precisely when its complement is countable. Equivalently, the closed sets are all subsets containing $a$ together with all countable subsets. Arbitrary intersections and finite unions preserve this latter condition, so it defines a topology.

Every convergent sequence in this space is eventually equal to its limit. Indeed the set of its values different from its limit is countable and closed; its complement is an open neighbourhood of the limit. Eventual membership in this complement forces eventual equality. Nevertheless $a$ is not isolated, because its singleton has uncountable complement. In particular sequence convergence alone does not determine this topology.

A real-valued function on this space is continuous exactly when $\{x:f(x)\ne f(a)\}$ is countable. Sufficiency follows by considering the preimages of closed subsets of the real line: each either contains $a$ or is a subset of that countable exceptional set. For necessity, each set $\{x:|f(x)-f(a)|\ge 1/n\}$ is closed and excludes $a$, hence is countable; their countable union is the exceptional set.

A Lindelöf space is a topological space in which every open cover has a countable subcover. A closed discrete subset of a Lindelöf space must be countable: cover each of its points by an open set meeting it at only that point and add the open complement. In a metrizable space, the Lindelöf property is equivalent to topological separability.

A finite nonempty Directed acyclic graph has a unique topological sort exactly when it has a Hamiltonian path. A Hamiltonian path forces the relative order of every consecutive pair and therefore of all vertices. Conversely, in a unique topological sort, every consecutive pair must have a forward edge: if such an edge were absent, swapping that adjacent pair would preserve all edge constraints and produce a second topological sort. The consecutive edges give a Hamiltonian path. Consequently one may compute any topological sort and test its consecutive edges, taking $O(n^2)$ time on an adjacency matrix representation.

An orientation of a topological manifold chooses a generator of every local homology group $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$, compatibly on coordinate neighbourhoods. Each choice is a local orientation of a manifold. Compatibility means the generators are locally restrictions of the same relative orientation class. It can be checked by choosing oriented coordinate balls and requiring agreement wherever their local classes overlap.

For a connected closed manifold, an orientation gives a unique fundamental class in $H_n(M;\mathbb{Z})$ whose restriction at each point is the chosen local generator. In an oriented triangulation this class is the sum of compatibly signed top simplices: interior faces cancel in opposite pairs, while the local coefficient in every top simplex is one. Reversing every local generator negates the fundamental class.

A reversing involution $R$ maps a trajectory $x(t)$ to another trajectory $Rx(-t)$. The chain rule gives the displayed condition. In contrast, an ordinary equivariance symmetry satisfies $F(Sx)=DS(x)F(x)$ and maps trajectories with time unchanged. Oddness $F(-x)=-F(x)$ expresses half-turn equivariance, not reversal by the half-turn. Damping usually breaks the mechanical candidate $R(x,y)=(x,-y)$.

A freely moving unstable particle of speed $v$ travels a laboratory distance $R$ in time $R/v$, accumulating proper time $R/(\gamma v)$. For rest-frame half-life $T$ and exponential decay, its survival probability is the displayed expression. At least half survive exactly when $\gamma vT\ge R$. The minimum speed is thereforeThe relevant flight length is $\gamma vT$, rather than $\gamma cT$ for a particle whose speed is appreciably less than $c$.

A computation-cost convention assigning constant cost to specified primitive operations. For exact unbounded integer arithmetic this can differ substantially from bit complexity: a multiplication of increasingly large values is one primitive operation but requires more bit operations. State which convention is used when estimating an algorithm's running time.

For every constant $c>0$, a deterministic multitape Turing machine with more than one tape and time complexity bounded by $T(n)$ has an equivalent machine with the same tape count and bound $T(n)/c+n+O(1)$, allowing a larger finite control and tape alphabet. Blocks of old symbols become single new symbols, and finite control implements blocks of old steps. The linear overhead accounts for initial input packing. If $T(n)/n\to \infty$, the overhead can be absorbed to obtain any fixed constant speed increase for sufficiently large inputs. This does not supply unlimited speed increases for operations on fixed hardware.^[ref]

A time complexity bounded by $2^{p(n)}$ for a polynomial $p$, where $n$ is the encoded input length. This usage includes bounds such as $2^{n^2}$; it need not mean only $2^{O(n)}$.

For isotropic linear elasticity with volumetric expansion coefficient $\alpha$, the stress-free thermal strain is $e^{\mathrm{th}}=\alpha (T-T_0)I/3$. Thus the thermal stress term is $-K\alpha (T-T_0)I$, where $K=\lambda +2\mu /3$ is the bulk modulus. A uniformly heated body with its full boundary fixed has zero displacement and the displayed compressive stress. For spatially varying temperature and zero body force, the displacement instead solves $\mu \Delta u+(\lambda +\mu )\nabla \mathrm{div}u=K\alpha \nabla T$ with its boundary values; the uniform formula is not automatic. A body force per volume $+K\alpha \nabla T$ cancels this thermal forcing in the equilibrium equation, but does not by itself cancel the stress or its boundary tractions.

Thermal strain is the strain that heating would produce without mechanical stress. In isotropic linear thermoelasticity it is a pure dilation: a volumetric expansion coefficient ${\alpha }_v$ corresponds to a linear expansion coefficient ${\alpha }_v/3$. The constitutive law acts on the mechanical strain $e-e^{\mathrm{th}}$. Spatially varying thermal strain need not be globally compatible with a stress-free displacement.

Thermal stress is stress generated when thermal expansion or contraction is constrained or incompatible. A uniformly heated isotropic body whose entire boundary is fixed has $e=0$ and $\tau =-K{\alpha }_v(T-T_0)I$. For a nonuniform temperature field, interior displacement generally remains nonzero even with a fixed outer boundary; equilibrium and compatibility must be solved together. Cancelling the divergence of a thermal stress term with a body force is not the same as making that stress vanish.

In a torsion-free divisible Abelian group, division by a nonzero integer is unique. The rational scalar operations are therefore a definitional expansion. Formulas reduce to Boolean combinations of rational linear equations: eliminate an existential variable in a conjunction by solving any equation with nonzero coefficient; if none fixes the variable, the finitely many nontrivial disequalities exclude only finitely many choices in an infinite group. Equations with zero coefficient reduce to parameter conditions.

Thus over a finite parameter set a complete one-variable type either specifies an element of its linear span, or excludes every element of that span. An uncountable rational vector space realizes the latter type because a finite-parameter span is countable. It is consequently an omega-saturated model in the additive language.

A temporal logic interpreted along infinite paths in a transition system. At position $i$, $X\alpha$ requires $\alpha$ at $i+1$, $F\alpha$ requires it at some $j\ge i$, and $G\alpha$ requires it at every $j\ge i$. Strong until and weak until express conditions that hold up to a later event. A system satisfies a specification when all relevant initial paths in a transition system satisfy it.

A branching temporal logic whose operators quantify over paths in a transition system from a state. In particular, $AG\Phi$, also written $\forall \square \Phi$, holds at $s$ when every infinite path in a transition system from $s$ satisfies $\Phi$ at every position, including its initial position.