I need visual examples of topological concepts I'm trying to understand basic concepts of topology, unfortunately I'm a very visual person, and as much documentation there is on how to come up with closed/open/clopen/etc. There are very few visual examples (using actual sets of integers, shapes, etc). So it's very hard for me to understand. Not everyone learns through text or mathematical definitions, so I figured this could help other people that have the same difficulties I have. Thanks for your help!
Can anyone get an example of a topology over a set.
I need one for a topology over a set, an open set, closed set, and clopen set. It needs to be something real. Like, say a set contained numbers { 1,2,3,4 }
It doesn't have to be that set, but it has to be a actual set of something (numbers, letters, geometric shapes), no labels (set X is the union of set Y and set Z), etc.
 A: Consider the set $X=\{1,2,3\}$.


*

*We have the trivial topology, namely $T=\{\emptyset,\{1,2,3\}\}$.

*We have the discrete topology, in which every singleton is open. This produces the topology to be $P(X)=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$

*We may have something in between declaring the nonempty open sets to be those containing $1$, and then we have $\{\emptyset,\{1\},\{1,2\},\{1,3\},\{1,2,3\}\}$.


Closed sets are those whose complement is open, and clopen sets are those which are open and closed as well.
Note that $X$ is always clopen relatively to $X$ (it might not be if we take a larger set and endow it with a different topology).
In the trivial topology we only have clopen sets, because we only require the minimum from the topology, while the set $\{1,2\}$ is neither open nor closed.
In the discrete topology every subset of $\{1,2,3\}$ is open, therefore every subset is closed. Since every subset is both open and closed, every subset is clopen.
In the last example note that all those that include $1$ are open, so those who do not include $1$ are closed. Since $1$ cannot be in both a set an its complement we have that there are no clopen sets except the empty set and $\{1,2,3\}$.
However it is important to remember that this is all relative and clopen, open and closed sets are only in this relationship with a specific topology and space.
Addendum: I feel that I should add a very basic explanation about topology.
Suppose we have an underlying set $X$. A topology on $X$ is a collection of subsets which includes the empty set as well $X$ itself. It is closed under unions and finite intersections.
The sets which are in the topology are called open, and their complements are called closed. A set which is both open and closed is usually called clopen.
A: If you're looking for visuals on topology, look no further than Wolfram MathWorld:

The blood types (O+, AB+, etc) form a topology under the power set topology: http://tumblr.com/xp12cy25v0. 
That's a visual for the power set topology on {1,2,3}. And as Asaf said, you can topologise {1,2,3} with the trivial topology or something else (as long as your $\cup$s and $\cap$s commute), though the above visual wouldn't work (nor would the logic).
Also @Xaan, a tip: you might want to be more polite, since people are helping you here for free.
A: This picture is from Topology , by J. Munkres.

A: Knowledge comes in the doing. So, in order to understand topological definitions, it would be wise to study a specific real-world example, and a great real-world example is the celebrated topological proof (by Furstenberg/פורסטנברג) of the infinitude of primes. Here’s the Wikipedia article about it.
Also, I realize that it is somewhat tangential to the thrust of your question, but I feel I should mention the classic book “Mathematical Snapshots” by Hugo Steinhaus. Here’s a review from the Amazon site:

Numerous photographs and diagrams help explain and illustrate mathematical phenomena in this series of thought-provoking expositions. Ranging from simple puzzles and games to more advanced problems, topics include the psychology of lottery players, the arrangement of chromosomes in a human cell, new and larger prime numbers, the fair division of a cake, how to find the shortest possible way to link a dozen locations by rail, and many other absorbing conundrums. A fascinating glimpse into the world of numbers and their uses. 1969 edition. 391 black-and-white illus.

Regards,
Mike Jones
