# Topological Spaces

• Gregory L. Naber
Part of the Texts in Applied Mathematics book series (TAM, volume 25)

## Abstract

We begin by recording a few items from real analysis (our canonical reference for this material is [Sp1], Chapters 1–3, which should be consulted for details as the need arises) . For any positive integer n, Euclidean n-space n = {(x 1 , ... , x n ) : x i ∊ ℝ. , i = 1, ..., n } is the set of all ordered n-tuples of real numbers with its usual vector space structure (x + y = (x 1, . . . , x n )+(y 1, . . . , y n ) = (x 1 + y 1, . . . , x n + y n ) and ax = a(x 1, ... , x n ) = (ax 1, ... , ax n )) and norm (||x|| = ((x 1)2+ ... +(x n )2)½). An open rectangle in ℝ n is a subset of the form (a 1 , b 1) × .... × (a n , b n ), where each (a i , b i ), i = 1,...., n, is an open interval in the real line ℝ . If r is a positive real number and p ∊ ℝ n , then the open ball of radius r about p is U r (p) = { x ∊ ℝ n : x - p < r}. A subset U of ℝ n is open in n if, for each pU, there exists an r > 0 such that U r (p) ∈ U (equivalently, if, for each pU, there exists an open rectangle R in ℝ n with pRU). The collection of all open subsets of ℝ n has the following properties: (a) The empty set Ø and all of ℝ n are both open in ℝ n . (b) If { U α : α ∊ A } is any collection of open sets in ℝ n (indexed by some set A), then the union U α∈A U α is also open in ℝ n . (c) If { U 1, ... , U k } is any finite collection of open sets in ℝ n , then the intersection U 1∩...∩ U k is also open in ℝ n .

## Keywords

Topological Space Topological Group Open Cover Hausdorff Space Relative Topology
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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