What is wrong with this proof?

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In summary, the conversation discusses proving that a countably infinite cartesian product of connected spaces is connected. The proof involves showing that a subspace of the product space is connected, as well as the union of all possible subspaces. This method relies on the assumption that the product of two connected spaces is connected, or that this result has been previously taught.
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JasonJo
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I'm trying to prove that a countably infinite cartesian product of connected spaces is connected.

Let X be a connected space and let Y be the countably infinite cartesian product of copies of X, and suppose Y is equipped with the product topology.

So suppose Y is not connected. Let A and B be two disjoint, nonempty, open subsets of Y s.t. A U B = Y. Since A and B are open sets in the product topology, they are both of the form U1 X U2 X ... where each Ui's are not X for only finitely many i and each Ui is open in X. So let k be the sup of the index's for A and B which Ui's is not a copy of X.

So A intersect B = (U1 intersect Y1) X ... X (Uk intersect YK) x X x X... = empty set.
Hence for some j less than k, Uj intersect Wj is empty. Hence Uj and Wj form a separation of X since A U B is Y, then Ui U Wi = X, and for Uj and Wj, they are disjoint and nonempty.

So where did I go wrong?
 
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  • #2
Here's how you prove this (and this proof works even if the product is not countable)

-Let a be a fixed point of X (the product space, indexed by I).
-Let K be any finite subset of I. Let Y be the subspace of X consisting of all points whose kth component is the kth component of a (k any element of K). Show that Y is connected.
-Show that the union Z of all the Y's is connected (they share a common point!).
-Show that X equals the closure of Z, and thus X is connected (remember that a point is in the closure of a set iff every neighbourhood of the point intersects the set).

This method works assuming that you already know how to prove that the product of 2 connected spaces is connected (and thus by induction a finite product of connected spaces is connected), or you can simply quote that result if it was already taught in class.
 
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Related to What is wrong with this proof?

1. What is a proof?

A proof is a logical argument that uses established principles and reasoning to demonstrate the validity of a statement or proposition.

2. What makes a proof incorrect?

A proof can be incorrect if it contains a logical fallacy, an error in reasoning, or if it does not follow the established rules and principles of the mathematical or scientific field it is attempting to prove.

3. How can I tell if a proof is wrong?

There are a few ways to tell if a proof is wrong. One way is to check for any logical fallacies or errors in reasoning. Another way is to compare the proof to established principles and rules in the field and see if it follows them correctly. Additionally, you can have someone else review the proof to get a fresh perspective.

4. Can a proof be partially correct?

Yes, a proof can be partially correct. It may contain some valid arguments and accurately follow the established principles, but still have some errors or gaps in the logic.

5. How can I fix a wrong proof?

The best way to fix a wrong proof is to identify the error or gap in logic and then correct it by using sound reasoning and following the established principles and rules of the field. You can also seek feedback from other experts in the field to help identify and correct any errors.

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