Topology/Basic Concepts Set Theory

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This chapter is meant to be a short, concise introduction to the basic set concepts used throughout this book. It is not meant to be a comprehensive text book on set theory, for that see elsewhere in Wikibooks. Rather, it will list the material that the reader should be familiar with, and showcase the notation used.

[edit] Unions, Intersections and Relations

The empty set is denoted by symbol \emptyset.  A (finite) set consisting of elements x_1, x_2, \ldots, x_n is denoted \{x_1, x_2, \ldots, x_n\}. It is a bit sloppy but common practice not to distinguish very strictly between a singleton set \ \{x\}  and its single element \ x.

Let A and B denote two sets. Then the union is denoted A\cup B, the intersection A\cap B and the difference A\setminus B. If every element in A also belongs to B, we say that A is a "subset" of B. In other words, \forall x\in A:x\in B is equivalent to A\subseteq B. A key property of these sets is that A = B if and only if A\subseteq B and B\subseteq A. If A\subseteq B and A \ne B then A is a proper subset of B, which we denote A\subsetneq B. We do not use the notation A\subset B, as the meaning varies between subset and proper subset in various sources.

Finite ordered sets (or n-tuples) are denoted (x_1, x_2,\ldots, x_n). For two ordered sets X=(x_1,x_2,\ldots,x_n) and Y=(y_1,y_2,\ldots,y_n), we have X = Y if and only if \forall n:x_n = y_n (compare with the statement above regarding equality of unordered sets).

Ordered sets can be defined in terms of unordered sets. For example, the ordered pair \ <x,y>  was defined by Kazimierz Kuratowski as <x,y> := \left\{\{x\}, \{x,y\}\right\}. It is an exercise in tedious but not difficult case checking to show that \ <x_1,y_1>=<x_2,y_2>  if and only if \ x_1 = x_2  and \ y_1 = y_2.

Now n-tuple is defined as follows:

(x_1, x_2,\ldots, x_n)\ :=\ \{<1,x_1>,<2,x_2>,\ldots,<n,x_n>\}

[edit] Exercises

If you can successfully answer the following two problems, you are ready to study topology! Please take the time to solve these problems.

1. Prove that the empty set is a subset of every set.

2a) Consider the set An = (-n,n) for each n in the set of natural numbers. Does the union over all An (for n in the set of natural numbers) equal R (the set of all real numbers)? Justify your answer.

b) Prove that no finite subcollection of this collection has the property that the union of this finite subcollection equals R. Once you study topology, you will see that this constitutes a proof that R is not compact.

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