Why is the infinite product of the discrete two point space with itself, a topological homogeneous space?
why is the infinite product of the discrete two point space with itself, a topological homogeneous space?
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general-topology
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2Think of the two point space as the cyclic group of order 2. Then the infinite product is also a group. You can check that the product topology makes it a topological group, so it's certainly homogeneous. – 2012-02-19
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1No need for groups: a 2 point discrete space is homogeneous. Any product of homogeneous spaces is homogeneous: work per coordinate and a product of homeomorphisms is a homeomorphism. Done. Products "create" homeogeneity too: $[0,1]^{\mathbb{N}}$ is homogeneous even when $[0,1]$ is not, and there are stronger theorems of this kind for zero-dimensional spaces too. – 2012-02-21