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W is connected

Since W is countable, we cannot hope to prove W is connected. We prove instead that

is connected. First we need some topological lemmas. Give

the discrete topology and

the product topology, as usual. If

is a finite vector of 0's and 1's, let

be the set of sequences in

which start with v. The following lemma is the key ingredient in the connectivity proof.

Lemma 4.1

Let Y be a topological space. Suppose is a continuous map such that

for all , and all . (Here v0 denotes the vector v with 0 appended, etc.) Then the image of f is path connected.

[Proof]

Let M be a topological space. Give the product topology and let the symmetric group act on by permuting the coordinates. The space , which parameterizes n-element multisets, can be given the quotient topology.

Lemma 4.2

If is connected, and the multiset is in A for some , then the subset of all coordinates of points in A is connected.

[Proof]

Theorem 4.1

is connected.

[Proof]

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