
§2.1. Topological Spaces Let X be a set. A family T of subsets of X is
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
Pre-AP Geometry 1
... Reasoning with Properties from Algebra • Determine if the equations are valid or invalid, and state which algebraic property is applied – (x + 2)(x + 2) = x2 + 4 – x3x3 = x6 – -(x + y) = x – y ...
... Reasoning with Properties from Algebra • Determine if the equations are valid or invalid, and state which algebraic property is applied – (x + 2)(x + 2) = x2 + 4 – x3x3 = x6 – -(x + y) = x – y ...
N - 陳光琦
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems for any branch of mathematics. - Predicate logic with function symbols, the “=” operator, and a few proof-building rules is sufficient for ...
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems for any branch of mathematics. - Predicate logic with function symbols, the “=” operator, and a few proof-building rules is sufficient for ...
A Tour of Formal Verification with Coq:Knuth`s Algorithm for Prime
... There is no relation between the length of a program and the difficulty of its proof of correctness. Very long programs performing elementary tasks could be trivial to prove correct, while short programs relying on some very deep properties could be much harder. Highly optimized programs usually bel ...
... There is no relation between the length of a program and the difficulty of its proof of correctness. Very long programs performing elementary tasks could be trivial to prove correct, while short programs relying on some very deep properties could be much harder. Highly optimized programs usually bel ...
A GENERALIZATION OF FIBONACCI FAR
... summands are s apart in index and opposite sign summands are d apart in index. We call such representations (s,d) far-difference representations, which we formally define below. Definition 1.11 ((s, d) far-difference representation). A sequence {an } has an (s, d) far-difference representation if ev ...
... summands are s apart in index and opposite sign summands are d apart in index. We call such representations (s,d) far-difference representations, which we formally define below. Definition 1.11 ((s, d) far-difference representation). A sequence {an } has an (s, d) far-difference representation if ev ...
A CELL COMPLEX IN NUMBER THEORY 1. Introduction Let M(n
... The system of squarefree numbers less than or equal to n and ordered by divisibility presents itself immediately as a simplicial complex. The same is not true for the larger system of all such numbers. However, a construction is known [BV] which produces a CW complex of a similar nature. The CW stru ...
... The system of squarefree numbers less than or equal to n and ordered by divisibility presents itself immediately as a simplicial complex. The same is not true for the larger system of all such numbers. However, a construction is known [BV] which produces a CW complex of a similar nature. The CW stru ...
Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.