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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: 125
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

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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