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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.

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See for instance A Little Example in bigi.org.uk/2007/11/02/topology-5-topological-spaces –  lhf May 20 '11 at 18:15
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@Xaade: It's not clear to me what you mean. Every set has both the discrete (every point is open) and the indiscrete (only open sets are the whole set and the empty set) topologies. I can't imagine any way "permutation" would be meaningful in these topologies. –  jd.r May 20 '11 at 18:50
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@Xaade: Nobody I know downvotes because they don't know how to answer, they downvote because they think it's not a good question. Your comment above about "permutation of every element" shows confusion on your part. While I didn't downvote, I'll note that your question lacks motivation (why do you "need" this); and is unclear (what does it mean for an example to be "real"? what is a "visual set"?) Taking the attitude of "if you downvoted me, it must be because you are too dumb to come up with an answer" is unlikely to get you many friends, or many good answers. And there are no variables –  Arturo Magidin May 20 '11 at 19:56
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I'm not entirely sure what the question is asking, but in Munkres' Topology when he defines a topology (in one of the first chapters after the set theory stuff) he has pictures of different topologies on a set with three points. It might be useful for you to think of some corresponding topologies for four points and ask someone who knows topology a bit better (or just run through the definition carefully!) to see if they satisfy the conditions to be a topology. –  james May 20 '11 at 20:00
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@Xaade: one thing which is totally missing from your question is motivation and background. The confluence of (i) wanting to know the rudiments of general topology and (ii) being unable/unwilling to read a standard introductory text is somewhat unusual. You say "Not everyone learns through text or mathematical definitions..." which is true, but at a certain point in their undergraduate career students of mathematics get trained to do just this. So I gather you're not a student of mathematics in the formal sense. That's fine, but then...who are you? –  Pete L. Clark May 20 '11 at 21:49

4 Answers 4

up vote 5 down vote accepted

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.

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This part I get lost on: "In the discrete topology everything is open, therefore everything is closed and thus everything is clopen." –  Lee Louviere May 20 '11 at 20:15
    
Xaade: I had tried to clarify that part a bit. –  Asaf Karagila May 20 '11 at 20:20
    
@Asaf : So, every member of the topology is automatically open? If it's complement exists, it is also closed. In the discrete case, every member is closed, because every member has their complement in the topology as well. Therefore all members are clopen. –  Lee Louviere May 20 '11 at 20:49
    
A set (topologically speaking) is not like a door. A door is either open, or it is closed; it cannot be both open and closed. A set can be both open and closed, neither open nor closed...one or the other...etc... Keep in mind that "not open" is not equivalent to closed, and "not closed" is not equivalent to open. And...its status as closed, open, clopen....etc is relative to the topological space with respect to which it is defined. –  amWhy May 20 '11 at 20:52
    
@Xaade: By definition, the "topology on the set" is the collection of all open sets, so by definition an element of $\tau$ is open, and if it is open then it is an element of $\tau$. The complement always exists, and by definition the complement of an open set is closed (in fact, "closed" means "its complement is open"). "Clopen" just means "both open and closed". –  Arturo Magidin May 20 '11 at 20:53

This picture is from Topology , by J. Munkres.

topologies on set with 3 elements

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If you're looking for visuals on topology, look no further than Wolfram MathWorld: topological drawings

The blood types (O+, AB+, etc) form a topology under the power set topology: http://tumblr.com/xp12cy25v0. blood types topology

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.

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+1, nice blog, thanks Lao Tzu –  Shuhao Cao May 20 '11 at 20:08

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

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