Topology?
Moderators: gmalivuk, Moderators General, Prelates
 KingLoser
 Obnoxious Distraction
 Posts: 458
 Joined: Mon Aug 11, 2008 9:04 pm UTC
 Location: Port Láirge, Éire
 Contact:
Re: Topology?
This is the obvious answer, but you could use computer network topologies to describe the math?
Re: Topology?
Well, I was hoping for something more purely mathematical, but thanks.

 Posts: 1459
 Joined: Fri Apr 20, 2007 3:27 pm UTC
 Location: The Tower of Flints. (Also known as: England.)
Re: Topology?
What general level are your audience at?
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
dubsola
dubsola
Re: Topology?
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
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
Re: Topology?
TacTics 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.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
Re: Topology?
Token wrote:TacTics 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 hourlong presentation. One and twodimensional space is something everyone is familiar with, so the OP should take advantage of that.
Re: Topology?
Familiar, yes, but that's no reason to lie to people about definitions.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
Re: Topology?
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 nonT2 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.
Topology is fun to talk about, since people who don't actually study it have never heard any of this.
Re: Topology?
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.
 skeptical scientist
 closedminded spiritualist
 Posts: 6142
 Joined: Tue Nov 28, 2006 6:09 am UTC
 Location: San Francisco
Re: Topology?
Token wrote:TacTics 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.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
 MartianInvader
 Posts: 804
 Joined: Sat Oct 27, 2007 5:51 pm UTC
Re: Topology?
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).
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
 Yakk
 Poster with most posts but no title.
 Posts: 11106
 Joined: Sat Jan 27, 2007 7:27 pm UTC
 Location: E pur si muove
Re: Topology?
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.
Just fold it down the diagonal and glue the sides that touch should work.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Topology?
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...
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
 Yakk
 Poster with most posts but no title.
 Posts: 11106
 Joined: Sat Jan 27, 2007 7:27 pm UTC
 Location: E pur si muove
Re: Topology?
Gluing all sides of a square to a single point is gluing a bunch of 2spaces to a 1space. Seems inelegant when you can do it without collapsing a dimension, which is what all of the other glue constructions required.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
 gmalivuk
 GNU Terry Pratchett
 Posts: 26596
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Topology?
Token wrote:TacTics 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."
Re: Topology?
"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.
This might be a good way to introduce topological spaces in general without scaring everyone away with the abstractness.
Non est salvatori salvator,
neque defensori dominus,
nec pater nec pater,
nihil supernum.
neque defensori dominus,
nec pater nec pater,
nihil supernum.
 skeptical scientist
 closedminded spiritualist
 Posts: 6142
 Joined: Tue Nov 28, 2006 6:09 am UTC
 Location: San Francisco
Re: Topology?
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."
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Who is online
Users browsing this forum: No registered users and 14 guests