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Topological basis
A topological basis generates a topology through carrier coverage and pointwise local intersection refinement.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Markov inequality
For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold.
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Pigeonhole principle
A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value.
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Euler's theorem in number theory
For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n.
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Subsequence
A subsequence selects terms by a strictly increasing natural-index map and preserves their original order.
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Orbit of a finite group action
The images of one point under all group elements form its orbit; all orbits partition the point carrier.
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Monotone bounded sequence convergence theorem
A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum.
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Subspace topology
For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y.
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Chebyshev inequality
For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2.
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Inclusion-exclusion principle
The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset.
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Unique prime factorization of integers
Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order.
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Finite directed graph
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.
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Stabilizer of a finite group action
The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup.
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Bolzano-Weierstrass theorem for real sequences
Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map.
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Compact subset
A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover.
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Law of total expectation
On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X].
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Binomial theorem
With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients.
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Mobius function
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.
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Context-free grammar
A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side.
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Orbit-stabilizer theorem
For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order.
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Mean value theorem
If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope.
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Continuous image of a compact set
If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y.
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Law of total variance
For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means.
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Recurrence relation
An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms.