Topology, Geometry, and Gauge Fields pp 27-100 | Cite as

# Topological Spaces

## 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** ℝ^{ 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** ℝ^{ n } if, for each *p* ∊ *U*, there exists an *r* > 0 such that *U* _{ r }(*p*) ∈ *U* (equivalently, if, for each *p* ∊ *U*, there exists an open rectangle *R* in ℝ^{ n } with *p* ∊ *R* ∈ *U*). 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## Preview

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