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Topic: Problem from Willard's _General Topology_
Replies: 8   Last Post: Jan 11, 2014 8:54 PM

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Michael F. Stemper

Posts: 108
Registered: 9/5/13
Problem from Willard's _General Topology_
Posted: Jan 5, 2014 12:09 PM
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I've started on _General Topology_ by Stephen Willard, and am having a
little difficulty with Problem 1D, "Cartesian Products".

Part 1 of this problem reads:
Provide an inductive definition of "the ordered n-tuple (x_1, ...,
x_n) of elements x_1, ..., x_n of a set" so that (x_1, ..., x_n) and
(y_1, ..., y_n) are equal iff their coordinates are equal in order,
i.e., iff x_1=y_1, ..., x_n=y_n.

My response to this makes use of the shorthand notation that
P_n = (x_1, ..., x_n). For instance, P_3 = (x_1, x_2, x_3).

I then used P_1 = { {x_1} } as the induction base of the requested
definition.

The induction step is then P_(n+1) = P_n U { (U P_n) U {x_(n+1)} }.

(I am using "U" here to represent the union symbol in both its unary
and binary incarnations.)

For n=1, these together give P_2 = { {x_1}, {x_1, x_2} }, which is the
definition of "ordered pair" that Willard uses.

For n=2, they give P_3 = { {x_1}, {x_1, x_2}, {x_1, x_2, x_3} }.

So far, so good. (I think.)

Part 2 of this problem asks us to define the Cartesian product of
an arbitrary number of sets, (a) using the definition of ordered
n-tuple from part 1, and (b) inductively from the definition of the
Cartesian product of two sets.

(a) is simple enough: X_1 * ... * X_n is the set of all ordered n-tuples
(x_1, ..., x_n) such that Ak (1<=k<=n) x_k in X_k.

(I'm using an asterisk to represent "product", since I already have "X"
and "x" in heavy use.)

(b) seems simple enough, as well. Define the trivial Cartesian product
of X_1 as { {x} | x in X_1 }. Then, let the Cartesian product X_1 * ... *
X_n * X_(n+1) be defined as the product (X_1 * ... * X_n) * X_(n+1).

All good so far (I think).

Now comes my difficulty, and I appreciate your patience in bearing with
me thus far. The two approaches that I've given are not "the same",
although they're equivalent.

For instance, the first approach gives:
P_3 = { {x_1}, {x_1, x_2}, {x_1, x_2, x_3} }, as I showed above.

However, the second approach gives:
P_3 = { { {x_1}, {x_1, x_2} }, { { {x_1}, {x_1, x_2} }, x_3 } }.

I see several possibilities:
a. One of the definitions that I developed is just plain wrong.
b. My definitions are okay, but I messed up when applying them.
c. My definitions are okay, but just not the same ones that Willard
assumed the student would come up with.
d. I'm over-interpreting the phrase "approaches are the same."

Is anybody willing to help me see which is the case, and give me some
hints on how to proceed?

Thanks much,
--
Michael F. Stemper
Always use apostrophe's and "quotation marks" properly.



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