topological dynamical system 拓扑动力系统
In the discrete-time convention, a pair $(X,T)$ with a topological space $X$ and continuous self-map $T:X\to X$, whose evolution is given by the iterates of $T$.
Terminology & Translations
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topological embedding 嵌入映射
A continuous map that is a homeomorphism onto its image equipped with the subspace topology.
topological entropy 拓扑熵
For continuous $T$ on a compact space, $h_{\mathrm{top}}(T)=\sup_{\mathcal U}\lim_{n\to\infty}n^{-1}\log N(\bigvee_{j=0}^{n-1}T^{-j}\mathcal U)$, where $\mathcal U$ ranges over finite open covers and $N$ is the least…
topological manifold 拓扑流形
For a fixed nonnegative integer n, a Hausdorff, second countable space in which each point has a neighborhood homeomorphic to an open subset of real n-dimensional Euclidean space; boundary is excluded in this record.
topological mixing 拓扑混合
A continuous self-map $T$ is topologically mixing when for every nonempty open $U,V$, some $N$ makes $T^n(U)\cap V\ne\varnothing$ for every $n\ge N$.
topological space 拓扑空间
A pair consisting of a set and a family of subsets containing the empty set and the whole set and closed under arbitrary unions and finite intersections.
topological transitivity 拓扑传递
A continuous self-map $T$ is topologically transitive if for every nonempty open $U,V$ some $n\ge0$ satisfies $T^n(U)\cap V\ne\varnothing$.
total differential 全微分
For f from an open subset of R^n to R^m, the total differential at a is the linear map A for which f(a+h)-f(a)-Ah=o(|h|), if such A exists.
trace of a number-field element 数域元素的迹
For a number field $K$ and $\alpha\in K$, $\operatorname{Tr}_{K/\mathbb Q}(\alpha)$ is the trace of the $\mathbb Q$-linear map $x\mapsto\alpha x$ on $K$.
tree in graph theory 树
A tree is a nonempty connected simple graph containing no cycle; this record restricts to finite trees.
two-color clique Ramsey number 拉姆齐数
For positive integers s,t, R(s,t) is the least positive N such that every red-blue edge coloring of K_N has a red K_s or a blue K_t.
uniform boundedness principle 一致有界原理
For a family of bounded linear maps from a Banach space X to a normed space Y, if sup_T norm(Tx) is finite for every x in X, then sup_T norm(T) is finite.
uniform continuity 一致连续
A map f between metric spaces is uniformly continuous if for every epsilon > 0 there is delta > 0 such that d(x,y) < delta implies d(f(x),f(y)) < epsilon for every pair of domain points.
uniform convergence 一致收敛
A sequence of real functions f_n on a common set E converges uniformly to f if for every epsilon > 0 some N makes |f_n(x)-f(x)| = N and all x in E.
union of sets 并集
The union of A and B contains exactly the objects belonging to at least one of them.
unique ergodicity 唯一遍历性
A continuous self-map of a nonempty compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure.
unique factorization domain 唯一分解环
A domain in which each nonzero nonunit factors into irreducibles uniquely up to factor order and multiplication by units.
unit of a ring 环的可逆元
An element $u$ in a unital ring admitting $v$ with $uv=vu=1$.
unit of an adjunction 单位
For $F\dashv G$, the natural transformation $\eta:1_C\Rightarrow GF$ whose component corresponds to $1_{FX}$ under the adjunction bijection.
unital ring 含幺环
An abelian additive group with associative multiplication, two-sided distributivity, and a multiplicative identity.
universal covering space 泛复迭空间
A covering with path connected and simply connected total space.
universal property 泛性质
A characterization of specified object-and-map data by a unique factorization for every competing compatible object-and-map datum.
universal quantifier 全称量词
$\forall x\,\varphi$ holds under an assignment in a structure precisely when $\varphi$ holds for every domain value assigned to $x$.
unramified extension of p-adic local fields p 进局部域的非分歧扩张
A finite extension $L/K$ of finite extensions of $\mathbb Q_p$ having ramification index $e(L/K)=1$.
van Kampen theorem Van Kampen 定理
For a space covered by two path connected open sets with nonempty path connected intersection and a basepoint in that intersection, its fundamental group is the pushout of the two fundamental groups over that of the…