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From: Sadeq on 2 Aug 2010 23:43
Recently, while reading a paper, I encountered the term "Elementary
Cylinder Set." (See http://en.wikipedia.org/wiki/Cylinder_set for more
info.)

Wandering about in several books related to Topology (as well as other
subjects in math, such as Set Theory, Probability Theory, Measure
Theory, etc.), I couldn't find a clue why such sets are called
"Cylinder Sets." What is the intuition behind? In addition, do we have
sets such as "Cone Sets," "Torus Sets," and so on?
From: William Elliot on 3 Aug 2010 03:58
On Mon, 2 Aug 2010, Sadeq wrote:

> Recently, while reading a paper, I encountered the term "Elementary
> Cylinder Set." (See http://en.wikipedia.org/wiki/Cylinder_set for more
> info.)

> Wandering about in several books related to Topology (as well as other
> subjects in math, such as Set Theory, Probability Theory, Measure
> Theory, etc.), I couldn't find a clue why such sets are called
> "Cylinder Sets." What is the intuition behind?

Within R^3, S = (0,1)x(0,1)xR is a solid square cylinder,
the geomertic interor of the square cylinder
.. . {0,1}x[0,1]xR \/ [0,1]x{0,1}xR.

S is homeomorphic to the solid cylinder
.. . { (x,y) | x^2 + y^2 < 1 } x R.

> In addition, do we have sets such as "Cone Sets,"
> "Torus Sets," and so on?
>
No. There is however, a topological generalization of the geometric cone.
From: Sadeq on 3 Aug 2010 07:13
On Aug 3, 11:58 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Mon, 2 Aug 2010, Sadeq wrote:
> > Recently, while reading a paper, I encountered the term "Elementary
> > Cylinder Set." (Seehttp://en.wikipedia.org/wiki/Cylinder_setfor more
> > info.)
> > Wandering about in several books related to Topology (as well as other
> > subjects in math, such as Set Theory, Probability Theory, Measure
> > Theory, etc.), I couldn't find a clue why such sets are called
> > "Cylinder Sets." What is the intuition behind?
>
> Within R^3, S = (0,1)x(0,1)xR is a solid square cylinder,
> the geomertic interor of the square cylinder
> . . {0,1}x[0,1]xR \/ [0,1]x{0,1}xR.
>
> S is homeomorphic to the solid cylinder
> . . { (x,y) | x^2 + y^2 < 1 } x R.
>
> > In addition, do we have sets such as "Cone Sets,"
> > "Torus Sets," and so on?
>
> No.  There is however, a topological generalization of the geometric cone.

Thanks a lot!
From: Dave L. Renfro on 3 Aug 2010 10:33
Sadeq wrote:

> Recently, while reading a paper, I encountered the term
> "Elementary Cylinder Set."
> (See http://en.wikipedia.org/wiki/Cylinder_set for more info.)
>
> Wandering about in several books related to Topology (as well
> as other subjects in math, such as Set Theory, Probability
> Theory, Measure Theory, etc.), I couldn't find a clue why
> such sets are called "Cylinder Sets." What is the intuition
> behind? In addition, do we have sets such as "Cone Sets,"
> "Torus Sets," and so on?

Looking at the definition, I see that an ordinary (infinite in
both directions) right circular cylinder is the inverse image
of a circle (1-sphere) under the second coordinate map of the
product space R x (R^2). You can get 3-dim rectangular plank
regions (2-by-4 planks like you use to build wooden houses with)
of infinite extent in the same way, except use planar rectangular
regions in the second factor space R^2 of R x (R^2). To get
rectangular planar strips parallel to a coordinate axes, use
the inverse images of bounded intervals under either coordinate
map in the product space R x R. I guess these all have a
"cylindrical feel" about them, in the sense that, with finite
dimensional Euclidean spaces as factors, you get regions that
have cross-sections all congruent in one or more directions.

There is also the notion of a "cone over a space" used in
algebraic topology:

http://en.wikipedia.org/wiki/Cone_(topology)

As for generalizations of torous sets, I would guess products
of spheres (especially used in homotopy theory) would be the
relevant generalization (recall that the ordinary torous is
the product space S^1 x S^1):

http://www.google.com/search?as_q=+homotopy&as_epq=products+of+spheres

Dave L. Renfro
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