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Getting close! Need to do limit points, then elegant definition of continuity, and that should do it.
Open intervals are not the only open sets. Any union of open intervals is an open set. For example:
Open intervals are not the only open sets. Any union of open intervals is an open set. For example:
::<math>\bigcup_{N \textrm{\ is\ an\ integer\ } \geq 2} (N, N+1/N)</math>
::<math>\bigcup_{N \textrm{\ is\ an\ integer\ } \geq 2} (N,N+1/N)</math>
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.
==Theorems==
==Theorems==
Here are a few extremely fundamental and far-reaching theorems:
Here are a few extremely fundamental and far-reaching theorems. Some of them are surprisingly simple:
Theorem: Neighborhoods are open sets.
Theorem: Neighborhoods are open sets.
::<math>\|x-C\| = k,\ \ \ k < r\,</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.
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.
Theorem: Any union of open sets, including unions of an infinite number of open sets, is an open set.
Proof: If a point <math>x</math> lies in the union, it must lie within one of the constituent open sets. There must be a neighborhood of <math>x</math> contained in that constituent open set. That neighborhood must be contained in the union.
Theorem: The intersection of two open sets is an open set.
Proof: Let <math>X = P_1 \cap P_2</math>, and let <math>x \in X</math>. Then <math>x \in P_1</math> and <math>x \in P_2</math>. Since <math>P_1</math> and <math>P_2</math> are open, there must be neighborhoods <math>N_1 \subseteq P_1</math> and <math>N_2 \subseteq P_2</math> that contain <math>x</math>. Whichever of those two neighborhoods has the smaller radius will be a subset of both <math>P_1</math> and <math>P_2</math>, so it will be a subset of <math>X</math>.
This theorem can be extended for any '''finite''' intersection, but it does not work for infinite intersections. Here is an example:
Let <math>P_i</math> be an infinite sequence of ever-decreasing open intervals:
::<math>P_i = \{ x\ |\ -1 - 1/i < 1+1/i \}\,</math>
for integer <math>i \ge 1</math>
The intersection of all of the <math>P_i</math>'s is the closed interval
::<math>[ -1, 1 ]\,</math>
which is not open.
So the topological rule of thumb is:
::'''Any''' union of open sets is open.
::Any '''finite''' intersection of open sets is open
Theorem: The null set (empty set) is open.
Proof: It needs to contain a neighborhood of each of its points. But it has no points.
Theorem: The entire space is open.
Proof: We need a neighborhood of each point in the space. The neighborhood centered on that point, with radius 1, will do the trick.
This means that the real line is open. It is '''not''' an open interval, because that interval would have to be "<math>(-\infty, \infty)</math>, and '''infinity is not a number'''. The real line is an open '''set''', because it is:
::<math>\mathbb{R}^1 = \bigcup_{n}\ (n-1, n+1)</math>
over all integers <math>n</math>. (Infinite unions are allowed, even though infinity is not a number.)
Theorem: Every open set is a union of neighborhoods.
Proof: It contains a neighborhood of each of its points; those are its constituent neighborhoods.
This means that any open set in the plane, for example, is a union of open circles. (It is also the union of open rectangles, open diamonds, open 5-pointed stars, and so on. This is a consequence of the invariance of the metric in defining a topology.)
In the field of topology, a collection of open sets, whose unions comprise all of the opens sets that exist, is called a '''basis'''. So what we have just shown is that the open neighborhoods (open intervals, open circles, open spheres, etc.) are a basis for the topology of finite-dimensional Euclidean spaces.
==Closed sets==
Definition: A set is '''closed''' if its complement is open.
That's all there is to it.
Because of some simple theorems of set theory, including DeMorgan's laws, some of the preceding theorems relating to open sets can be reformulated for closed sets.
Any intersection of closed sets, including the intersection of an infinite number of closed sets, is closed.
Any union of a finite number of closed sets is closed.
The null set is closed.
The entire space (for example, the real line) is closed.
==Limit points, and the other definition of closed sets==
Closed sets are sometimes given a different definition, as sets containing their limit points.
[[category:mathematics]]
[[category:mathematics]]