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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.
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.
Theorem 4.1
is connected.
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