Connectedness is not box product-closed
This article gives the statement, and possibly proof, of a topological space property (i.e., connected space) not satisfying a topological space metaproperty (i.e., box product-closed property of topological spaces).
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Intuitively, the idea is that points that are far away from each other on infinitely many coordinates (as opposed to finitely many coordinates) are in different connected components.
Proof using a countable power of the real line
Set each with the usual Euclidean space topology and suppose . Thus, the underlying set of is the set of all sequences of real numbers.
can be represented as a union of two nonempty disjoint open subsets and , where:
- is the set of bounded sequences, i.e., sequences for which there exists a real number satisfying for all .
- is the set of bounded sequences, i.e., sequences for which there exists no real number satisfying for all .
To see that both and are open, note that if , then all points in an open box of finite radius in all dimensions about is also in . The corresponding statement is true for .
Note that this proof crucially depends on the box topology, where it is possible to choose boxes that do not span the whole space in all (infinitely many) dimensions. These boxes would not be open in the product topology.