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More, still a work in progress, obviously.
::<math>\{ x\ |\ \|x-C\| \le r \}\,</math>
::<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.)
and the picture would be a solid circle. But open neighborhoods are the important sets from a theoretical standpoint.)
==Definition of open set==
Here is the proper theoretical definition:
Definition: A set is '''open''' if it contains a neighborhood of each of its points.
What this means is that a set <math>X</math> is open if, for every point <math>x</math> in <math>X</math>, there is a neighborhood <math>N</math> such that <math>x \in N</math> and <math>N \subseteq X</math>. This construction is shown in the following diagram:
::::''Need a picture here!''
If the point <math>x</math> were allowed to lie exactly on the edge of <math>X</math>, it wouldn't be possible to draw a nonzero neighborhood around <math>x</math> that lies in <math>X</math>. So the important feature of <math>X</math>'s openness is that no point can lie exactly on its edge. Every point in <math>X</math> must be some finite distance back from the edge, which makes it possible to draw a neighborhood around it.
==Theorems==
Here are a few extremely fundamental and far-reaching theorems:
Theorem: Neighborhoods are open sets.
Proof: Suppose a neighborhood has center <math>C</math> and radius <math>r</math>. If a point <math>x</math> is in that neighborhood, its distance from <math>C</math> must be strictly less than <math>r</math>, call it <math>k</math>.
::<math>\|x-C\| = k,\ \ \ k < r\,</math>
Place a new neighborhood, of radius <math>(r-k)/2</math>, around <math>x</math>. Every point in that neighborhood has a distance less than <math>k + (r-k)/2</math> from <math>C</math>. That distance is less than <math>r</math>, so every point in the new neighborhood is in the original neighborhood, so the new neighborhood lies within the original one.
[[category:mathematics]]
[[category:mathematics]]