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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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