A simpler construction of a peano space

Date

10-1993

Degree

Bachelor of Science in Applied Mathematics

College

College of Arts and Sciences (CAS)

Adviser/Committee Chair

Edgar E. Escultura

Abstract

This paper is concerned with the construction of the Peano space-filling continuous map of the unit interval I=[0,1] onto the unit square 1 x 1. 1 he unit interval was subdivided and mapped each subinterval suitably into each subsquare. The process was iterated until it completely fills the unit square.

The sequence of continuous functions f1{t}, f2{t}…fn{t} was generated recursively using the method of rotation, translation and contraction. A definition: f(x) is uniformly continuous if for every c>0, there exists a 8 > 0 (depending only one) such that if x, y E S and [x - y = 6, then I f(x) f(y) < e lemma: a function which is continuous on a compact set is uniformly continuous; and a well-known theorem: the limit of a sequence of uniformly continuous function is continuous were used to construct a continuous map of I onto the !milt square. The unit square is not empty since it was shown that the preimage of every point of I x I is covered by the limit map. The cardinality of the preimage in either corner point or interior point is countable. This makes the density of the preimage countable by the definition that density Is the cardinality of the preimage of each point. The distribution of the density is uniform. The construction of a Peano space is simpler than the other constructions which have appeared in print because you can find a generator such that every function at every stage can be expressed as a finite number of translations and contractions and since this generator can be expressed explicitly as a function, then every function in the sequence can be explicitly expressed by the generator.

Language

English

Location

UPLB Main Library Special Collections Section (USCS)

Call Number

Thesis

Document Type

Thesis

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