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MAT425 Real Analysis L7

Prof. Thistleton

Neighborhoods We often consider not just whether an individual point on the real line has some characteristic, but also whether those points near share the characteristic.

We may think of an interval as the set of points between two real numbers and write the interval as (a,b). We may also consider an interval to be the set of points within a certain distance of the center of the interval. The following theorem makes this idea more precise.

Theorem 110

If a < b, let c = (a+b)/2 and $\epsilon = (b-a)/2$ .Then $(a,b) = \{x: \vert x-c\vert < \epsilon\}$ .

The interval which has a point c as its center is especially useful in our work and so we give it a name.

Definition 116

An interval of the form $\{x: \vert x-c\vert < \epsilon\}$ for some real number c and positive real number x is called an epsilon neighborhood of c.

Generalize the idea of an epsilon neighborhood of a point c by considering any set which contains c and at least some epsilon neighborhood around c.

Definition 125

The set U is a neighborhood of the point c if there exists some $\epsilon_0 \gt 0$ such that U contains this epsilon neighborhood of c.

Consider an open interval (a,b). Let $x \in (a,b)$ . We can show that there is an epsilon neighborhood of x wholly within (a,b). This is a property which will later distinguish what we call open sets.

Theorem 130

An open interval is a neighborhood of each of its points.

If two real numbers are distinct, then we may separate them with neighborhoods.

Theorem 137

Let $x, y \in \Re, x \ne y$ . Show $\exists$ neighborhoods U, V such that $U \cap V = \emptyset$ and $x \in U$ and $y \in V$ .

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William Thistleton
11/10/1998