# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=9#results)[1](/en/knowledge/?page=1#results) … [9](/en/knowledge/?page=9#results) 10 [11](/en/knowledge/?page=11#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=11#results) 302 results · Page 10 of 13 Knowledge object ## [Topological basis](/en/mko/mko-topological-basis/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) A topological basis generates a topology through carrier coverage and pointwise local intersection refinement. mko-topological-basis · Topology Knowledge object ## [Markov inequality](/en/mko/mko-markov-inequality/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold. mko-markov-inequality · Probability and statistics Knowledge object ## [Pigeonhole principle](/en/mko/mko-pigeonhole-principle/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value. mko-pigeonhole-principle · Combinatorics and discrete mathematics Knowledge object ## [Euler's theorem in number theory](/en/mko/mko-euler-theorem-number-theory/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n. mko-euler-theorem-number-theory · Number theory Knowledge object ## [Subsequence](/en/mko/mko-subsequence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) A subsequence selects terms by a strictly increasing natural-index map and preserves their original order. mko-subsequence · Analysis Knowledge object ## [Orbit of a finite group action](/en/mko/mko-finite-group-orbit/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) The images of one point under all group elements form its orbit; all orbits partition the point carrier. mko-finite-group-orbit · Algebra Knowledge object ## [Monotone bounded sequence convergence theorem](/en/mko/mko-monotone-sequence-convergence-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum. mko-monotone-sequence-convergence-theorem · Analysis Knowledge object ## [Subspace topology](/en/mko/mko-subspace-topology/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y. mko-subspace-topology · Topology Knowledge object ## [Chebyshev inequality](/en/mko/mko-chebyshev-inequality/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2. mko-chebyshev-inequality · Probability and statistics Knowledge object ## [Inclusion-exclusion principle](/en/mko/mko-inclusion-exclusion-principle/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset. mko-inclusion-exclusion-principle · Combinatorics and discrete mathematics Knowledge object ## [Unique prime factorization of integers](/en/mko/mko-unique-prime-factorization/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order. mko-unique-prime-factorization · Number theory Knowledge object ## [Finite directed graph](/en/mko/mko-finite-directed-graph/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges. mko-finite-directed-graph · Combinatorics and discrete mathematics Knowledge object ## [Stabilizer of a finite group action](/en/mko/mko-finite-group-stabilizer/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup. mko-finite-group-stabilizer · Algebra Knowledge object ## [Bolzano-Weierstrass theorem for real sequences](/en/mko/mko-bolzano-weierstrass-theorem-real/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map. mko-bolzano-weierstrass-theorem-real · Analysis Knowledge object ## [Compact subset](/en/mko/mko-compact-subset/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover. mko-compact-subset · Topology Knowledge object ## [Law of total expectation](/en/mko/mko-law-of-total-expectation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]. mko-law-of-total-expectation · Probability and statistics Knowledge object ## [Binomial theorem](/en/mko/mko-binomial-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients. mko-binomial-theorem · Combinatorics and discrete mathematics Knowledge object ## [Mobius function](/en/mko/mko-mobius-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign. mko-mobius-function · Number theory Knowledge object ## [Context-free grammar](/en/mko/mko-context-free-grammar/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side. mko-context-free-grammar · Mathematics of computer science Knowledge object ## [Orbit-stabilizer theorem](/en/mko/mko-orbit-stabilizer-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order. mko-orbit-stabilizer-theorem · Algebra Knowledge object ## [Mean value theorem](/en/mko/mko-mean-value-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D10) If a