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https://www.math.tu-dresden.de/~bodirsky/lehre/alg-strukturen-ss-2016/script.pdf

Consider $Sym( \mathbb{N})$ as a topological group.

Let $P$ be the set of permutations $f$ of $\mathbb{N}$ that have finite support, i.e. $|\{i \in \mathbb{N} | f(i) \neq i\} |$ is finite.

In this lecture notes, Example 1.4.2 says that the closure of $P$ is the whole symmetric group.

Here a set $S$ is closed if it contains all $f \in Sym(\mathbb{N})$ such that for all finite $F \subset \mathbb{N}$ there exists $g \in S$ such that $f(x)=g(x) \forall x \in F$.

So my question is how do you show that $P$ is not closed and that the closure of $P$ is the whole group.

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    Hey, why don't accept the good answers you got? This is the appropriate way to thank people who answered your questions. – 2017-01-15

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Consider a non-empty basic open set $S(\bar a, \bar b)$ of $\operatorname{Sym}(\mathbb{N})$, where $\bar a = (a_{1}, \dots, a_{k})$ and $\bar b = (b_{1}, \dots, b_{k})$ for some $k$. Consider $$ \Delta = \{ a_{1}, \dots, a_{k}, b_{1}, \dots, b_{k} \} $$ and the map $\sigma$ which is defined as

  • first of all as $\sigma(a_{i}) = b_{i}$ for $i = 1, \dots, k$,
  • then extended to a bijection of $\Delta$ (one can do this because as $S(\bar a, \bar b)$ is non-empty, we have $b_{i} \ne b_{j}$ for $a_{i} \ne a_{j}$), and
  • finally extended to the identity on $\mathbb{N} \setminus \Delta$.

Then $\sigma \in S(\bar a, \bar b)$ has support $\subseteq \Delta$.

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    Could you explain a little bit more how that explains the closure? I have added the definition of closure in the question. Thanks. – 2017-01-14
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    I am using the definition of closure for a generic topological space. $a$ is in the closure of $B$ if any open set $U$ containing $a$ intersects $B$ non-trivially. In this case, I show that any non-empty open set intersects $P$ non-trivially, so the closure of $P$ is the whole space. You can check that the definition I used coincides with yours in this particular case. And I am using the notation of your notes. – 2017-01-14