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→Open sets: More material -- neighborhoods in arbitrary dimension
Before giving the definition of open sets in Euclidean space, we present some examples. Readers who are aware of the general intuitive notion of open sets should find these examples familiar.
Before giving the definition of open sets in Euclidean space, we present some examples. Readers who are aware of the general intuitive notion of open sets should find these examples familiar.
===In one dimension===
The simplest open sets in 1-dimensional Euclidean space (formally called <math>\mathbb{R}^1</math>; informally called the real numbers of the "real line") are '''open intervals'''. An open interval consists of those numbers lying strictly between two endpoints a and b. In set-theoretic notation:
The simplest open sets in 1-dimensional Euclidean space (formally called <math>\mathbb{R}^1</math>; informally called the real numbers of the "real line") are '''open intervals'''. An open interval consists of those numbers lying strictly between two endpoints a and b. In set-theoretic notation:
Bizarrely defined sets like the one above are commonly used as examples and counterexamples in analysis and topology.
Bizarrely defined sets like the one above are commonly used as examples and counterexamples in analysis and topology.
===In two or more dimensions===
In two or more dimensions the situation becomes more complicated, because even simple open sets can come in an endless variety of shapes. The fundamental open set (equivalent to an open interval) is the '''open neighborhood''', also called an open ball. An open neighborhood has a center point and a nonzero radius, and is the set of all points whose distance from the center is strictly less than that radius. In set-theoretic notation:
::<math>\{ x\ |\ \|x-C\| < r \}\,</math>
The double-stroke absolute value sign is the '''norm''' or the '''metric distance function'''. In the common case it is the Euclidean/Pythagorean distance:
::<math>\|a-b\| = \sqrt{(a_1-b_1)^2 + (a_2-b_2)^2}\,</math> in two dimensions (similarly for higher dimensions)
The double-stroke absolute value sign is similar to the usual absolute value operation, generalized to arbitrary dimensions or other metric spaces.
It is easy to see that, in two dimensions, an open neighborhood is the interior of a circle. It does '''not''' include the actual boundary of the circle, because it consists of the points whose distance from <math>C</math> is '''strictly less''' than <math>r</math>. This point is crucial—the whole subject of analysis and topology depends on it!
To draw a picture of an open neighborhood, use a circle bounded by a dotted line:
::::''Need a picture here!''
(To make a closed ball, the formula would be:
::<math>\{ x\ |\ \|x-C\| \le r \}\,</math>
and the picture would be a solid circle. But open neighborhoods are the important sets from a theoretical standpoint.)
[[category:mathematics]]
[[category:mathematics]]