Topology?

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Dwalin
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Topology?

Postby Dwalin » Mon Oct 27, 2008 7:30 pm UTC

Any body know some interesting subjects in topology for an hour presentation, based on minimal prior knowledge?

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Re: Topology?

Postby KingLoser » Mon Oct 27, 2008 7:40 pm UTC

This is the obvious answer, but you could use computer network topologies to describe the math?
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Dwalin
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Re: Topology?

Postby Dwalin » Mon Oct 27, 2008 8:04 pm UTC

Well, I was hoping for something more purely mathematical, but thanks.

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Re: Topology?

Postby Ended » Mon Oct 27, 2008 8:21 pm UTC

What general level are your audience at?
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Re: Topology?

Postby Tac-Tics » Mon Oct 27, 2008 8:30 pm UTC

What level class are we talking about?

I think it might be good to show using simple, concrete examples. First, explain what open and closed sets are, in nontechnical terms, and give examples:

"An open set is a set where every point can be wiggled a little bit and still be inside the set

A closed set is a set whose compliment is open

Examples of open sets in R include (0, 1), (-pi, pi), and (0, infinity)

Examples of closed sets in R include [0, 1], [0, 42], [-10, 10]

Open and closed are NOT opposites. The interval (0, 1] is an example of a set which is NEITHER open NOR closed. The empty set and the set R itself are BOTH open AND closed.

Go on to explain these concepts generalize into R^2 naturally. A circle including all points inside and including the boundary is an example of a closed set. A circle including all its interior points, but without the boundary is an example of an open set.

An open set containing a point p is called an (open) neighborhood of p.

Then, it is nice to produce a useful result with the definitions. The topological definition of a function is a really good example.

A function f maps points from R^2 to R^2. We can also think of f mapping subsets of R^2 to subsets of R^2 in a natural way: If X is a set, then f(X) is the set {f(x) for x in X}.

Then, finally, a continuous function from R^2 to R^2 is one where the following is possible. Given any open set in R^2 called Y, we can find an open set in R^2 called X such that f(X) is a subset of Y.

The reason for using R^2 is it makes for a nicer picture. Wikipedia has a nice picture of what I mean (though, the names of the variables are different):

http://en.wikipedia.org/wiki/Continuous_function_%28topology%29
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Re: Topology?

Postby Token » Mon Oct 27, 2008 8:51 pm UTC

Tac-Tics wrote: "An open set is a set where every point can be wiggled a little bit and still be inside the set

That's the definition of an open set in a metric space, not a topological one.
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Re: Topology?

Postby Tac-Tics » Mon Oct 27, 2008 8:58 pm UTC

Token wrote:
Tac-Tics wrote: "An open set is a set where every point can be wiggled a little bit and still be inside the set

That's the definition of an open set in a metric space, not a topological one.


Like I said, start with a concrete example. We don't need to talk about the general case of a topological space, and it would be pointless to try to do so in an hour-long presentation. One- and two-dimensional space is something everyone is familiar with, so the OP should take advantage of that.

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Re: Topology?

Postby Token » Mon Oct 27, 2008 9:05 pm UTC

Familiar, yes, but that's no reason to lie to people about definitions.
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Re: Topology?

Postby Harg » Mon Oct 27, 2008 9:15 pm UTC

If your audience is at a high enough level, it might be fun to show some "strange" examples. Define a homeomorphism and talk about why we require a continuous inverse to a continuous mapping (ask them for an example; if they've only ever seen real functions of a real variable, they'll have to think about it), show them some homeomorphic spaces (specifically do the stereographic projection of a sphere onto a plane), maybe show some manifolds (they look nice; it might be worth it to mention fundamental polygons here, at least for the torus and the Möbius strip, perhaps the projective plane). Finally, some pathology: you could display the multitude of limits of a sequence in a non-T2 space etc. If you really want to, you can talk intuitively about the long line (although I doubt there's anything intuitive about it :))
Topology is fun to talk about, since people who don't actually study it have never heard any of this.

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Re: Topology?

Postby Tac-Tics » Mon Oct 27, 2008 9:29 pm UTC

Token wrote:Familiar, yes, but that's no reason to lie to people about definitions.

How it is a lie? It's a restriction of the standard definition.

If you wanted to go into detail and explain what a topology is... "there's a set X and a collection of subsets of X called T such that {} is in T and X is in T and for any subcollection U of T, the union of all sets in U are in T and for any finite subcollection U of T, the intersection is in T".... the students are going to miss the whole point. The entire concept of open and closed sets comes directly from observations in the metrics on R and R^2.

Skimp the technical stuff for conceptual understanding. You can't teach topology in an hour, but you can get the students interested in it in that time frame.

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Re: Topology?

Postby skeptical scientist » Tue Oct 28, 2008 2:01 am UTC

Token wrote:
Tac-Tics wrote: "An open set is a set where every point can be wiggled a little bit and still be inside the set

That's the definition of an open set in a metric space, not a topological one.

That depends what you mean by "a little it". If you mean "a distance smaller than epsilon", you have the metric space definition. If you mean "inside an element of some basis", then you have a general topological space definition.
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Re: Topology?

Postby MartianInvader » Tue Oct 28, 2008 2:42 am UTC

The intro I like to talk about is gluing together edges of a square in order to make a sphere (all glued into a single point), a torus (opposite sides glued), a klein bottle (one pair of opposite sides glued with a flip), and real projective space (both opposite sides glued with flips).
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Re: Topology?

Postby Yakk » Tue Oct 28, 2008 3:24 pm UTC

You don't need to glue them all to a single point to make a sphere.

Just fold it down the diagonal and glue the sides that touch should work.
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Re: Topology?

Postby Token » Tue Oct 28, 2008 4:00 pm UTC

Yakk wrote:You don't need to glue them all to a single point to make a sphere.

Just fold it down the diagonal and glue the sides that touch should work.

Why "you don't need to"? Your method is the more complicated of the two...
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Re: Topology?

Postby Yakk » Tue Oct 28, 2008 4:18 pm UTC

Gluing all sides of a square to a single point is gluing a bunch of 2-spaces to a 1-space. Seems inelegant when you can do it without collapsing a dimension, which is what all of the other glue constructions required. :)
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Re: Topology?

Postby gmalivuk » Tue Oct 28, 2008 10:33 pm UTC

Token wrote:
Tac-Tics wrote: "An open set is a set where every point can be wiggled a little bit and still be inside the set

That's the definition of an open set in a metric space, not a topological one.

You could simply throw in a "for example" if you're all butthurt about "lying" to people.

"A topology is a set along with a collection of open subsets of that set. In what's called a metric space, for example, an open set is one where every point inside has a small region around it that's still inside the set."
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Re: Topology?

Postby SunAvatar » Tue Oct 28, 2008 11:13 pm UTC

"A topology on a set is just a sort of rule telling us which points are 'close together'. It does this by specifying something called a closure operation: the closure of a set A is the set of all the points 'close' to A. Since the idea of topology is to study this idea in general, though, we'll let any operation count as a closure operation, and just redefine 'close' to match, as long as it satisfies a few basic rules: (list the Kuratowski closure axioms)"

This might be a good way to introduce topological spaces in general without scaring everyone away with the abstractness.
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Re: Topology?

Postby skeptical scientist » Wed Oct 29, 2008 5:31 am UTC

Who defines topological spaces using closures, anyways? I've only seen the open set definition. Also, calling the closure of A the set of points "close" to A can be confusing, since close is often used to mean "in some neighborhood of."
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