connected graph 连通图
A nonempty finite simple graph is connected when each pair of vertices is joined by a path.
Terminology & Translations
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connected space 连通
A space with no decomposition into two disjoint nonempty open subsets; the empty space is included here.
continuous dual space 对偶空间
The continuous dual X* of a normed space X is the vector space of bounded linear scalar-valued functionals on X, equipped with the operator norm.
continuous function 连续函数
A map between metric spaces is continuous if, at every domain point, every positive output tolerance is met by some positive input tolerance.
continuous uniform distribution 均匀分布
For real $a<b$, the law with density $1/(b-a)$ on $[a,b]$ and zero elsewhere.
continuum hypothesis 连续统假设
In ZFC, the assertion $2^{\aleph_0}=\aleph_1$, where $\aleph_1$ is the least uncountable cardinal.
contractible space 可缩空间
A nonempty topological space whose identity map is homotopic to a constant map.
contravariant functor 反变函子
A functor Cᵒᵖ→D, viewed as an assignment from C reversing the direction of arrows.
convergence in distribution 依分布收敛
Real-valued laws converge in distribution when their CDFs converge to the limiting CDF at all its continuity points.
convergence in probability 依概率收敛
Variables on one probability space satisfy $X_n\to X$ in probability when $P(|X_n-X|>\varepsilon)\to0$ for every $\varepsilon>0$.
convolution of functions 卷积
For f,g in L1(R^n), their convolution is the almost-everywhere-defined integrable function (f*g)(x)=integral f(x-y)g(y) dy.
convolution of sequences 卷积
For sequences (a_n) and (b_n) in a commutative ring indexed by n ≥ 0, their convolution is c_n = ∑_{k=0}^n a_k b_{n−k}.
coprimality of integers 整数互素
Integers $a,b$ are coprime when $\gcd(a,b)=1$.
cotangent space 余切空间
For a point p of a smooth real manifold M, the real dual vector space of its tangent space, consisting of real linear functions on tangent vectors at p.
counit of an adjunction 余单位
For $F\dashv G$, the natural transformation $\varepsilon:FG\Rightarrow1_D$ whose component corresponds to $1_{GY}$ under the inverse adjunction bijection.
countable set 可数集
A set that is finite or admits a bijection with the natural numbers, including the empty set.
counterexample 反例
An admissible instance satisfying a claim’s hypotheses but falsifying its asserted conclusion.
covariance 协方差
For square-integrable real $X,Y$, $\operatorname{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]$.
covering space 覆盖空间
A space E with a continuous surjection to B such that each point of B has a neighborhood whose inverse image splits into disjoint open sheets each homeomorphic to that neighborhood.
cumulative distribution function 分布函数
For a real random variable $X$, $F_X(x)=P(X\le x)$; it is nondecreasing, right-continuous, and has limits $0$ and $1$ at the two infinities.
cup product 杯积
The product on singular cohomology with coefficients in a commutative ring, induced by evaluating a p-cochain on the first p-face and a q-cochain on the last q-face of an ordered (p+q)-simplex and multiplying.
CW complex CW-复形
A space built by attaching disks to successive skeleta along their boundary spheres and giving the union the weak topology determined by the skeleta.
cycle in a simple graph 圈
In a simple graph, a cycle consists of distinct vertices v_1,…,v_k with k ≥ 3, consecutive pairs joined, and v_k joined to v_1; its length is k.
cyclic group 循环群
A group generated by one element, so every element is an integer power of that generator.
cyclotomic number field 分圆域
A field $\mathbb Q(\zeta_n)$ generated by a complex root of unity $\zeta_n$ of exact order $n$.