User:Caliburn/s/nets/Point is Cluster Point of Moore-Smith Sequence iff Limit of Monotonically Indexed Moore-Smith Subsequence

Theorem

Let $\struct {X, \tau}$ be a topological space.

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $\family {x_\lambda}_{\lambda \in \Lambda}$ be a Moore-Smith sequence.

Let $y \in X$.

Then $y$ is a cluster point of $\family {x_\lambda}_{\lambda \in \Lambda}$ if and only if:

there exists:

a directed set $\struct {\Omega, \le}$

a monotone cofinal mapping $\phi : \Omega \to \Lambda$

such that the Moore-Smith sequence $\family {x_{\map \phi m} }_{m \in \Omega}$ converging to $y$.

That is, $y$ is a cluster point of $\family {x_\lambda}_{\lambda \in \Lambda}$ if and only if some monotonically indexed Moore-Smith subsequence of $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $y$.

Proof

Necessary Condition

Let $y$ be a cluster point of $\family {x_\lambda}_{\lambda \in \Lambda}$.

Let:

$\Omega = \set {\tuple {\lambda, U} : \lambda \in \Lambda, \, U \text { is a neighborhood of } y \text { such that } x_\lambda \in U}$

Define a relation $\le$ on $\Omega$ by:

$\tuple {\lambda_1, U_1} \le \tuple {\lambda_2, U_2}$

if and only if:

$\lambda_1 \preceq \lambda_2$

and:

$U_1 \supseteq U_2$

We show that $\struct {\Omega, \le}$ is a directed set.

Define:

$M' = \set {\tuple {\lambda, U} : \lambda \in \Lambda, \, U \text { is an open neighborhood of } y}$

Let $\le'$ be the relation defined by:

$\tuple {\lambda, U} \le' \tuple {\lambda', U'}$ if and only if:

$\lambda \preceq \lambda'$ and $U \supseteq U'$.

From Product of Directed Sets is Directed Set and Restriction of Preodering is Preordering, we have that $\le$ is a preordering on $\Omega$.

We just need to show that $\le$ is a directed preordering on $\Omega$.

Let $\tuple {\lambda, U}, \tuple {\lambda', V} \in \Omega$.

Then since $U$ and $V$ are open neighborhoods of $y$, so is $U \cap V$.

Since $\struct {\Lambda, \preceq}$ is directed, there exists $\mu \in \Lambda$ such that:

$\lambda \preceq \mu$

and:

$\lambda' \preceq \mu$

Since $y$ is a cluster point of $\family {x_\lambda}_{\lambda \in \Lambda}$, there exists $\lambda_\ast \in \Lambda$ such that $\mu \preceq \lambda_\ast$ and $x_{\lambda_\ast} \in U \cap V$.

From transitivity, we have that $\lambda \preceq \lambda_\ast$ and $\lambda' \preceq \lambda_\ast$.

So we have $\tuple {\lambda, U} \le \tuple {\lambda_\ast, U \cap V}$ and $\tuple {\lambda', V} \le \tuple {\lambda_\ast, U \cap V}$.

So $\struct {\Omega, \le}$ is indeed directed.

Now $\phi : \Omega \to \Lambda$ by:

$\map \phi {\lambda, u} = \lambda$

for each $\tuple {\lambda, u} \in \Omega$.

We show that $\phi : \struct {\Omega, \le} \to \struct {\Lambda, \preceq}$ is monotone.

Suppose that $\tuple {\lambda_1, u_1} \le \tuple {\lambda_2, u_2}$.

Then $\lambda_1 \preceq \lambda_2$.

In particular, $\map \phi {\lambda_1, u_1} \preceq \map \phi {\lambda_2, u_2}$ whenever $\lambda_1 \preceq \lambda_2$.

So $\phi$ is monotone.

We show that $\phi$ is cofinal.

Let $\lambda \in \Lambda$.

Then, if $\tuple {\lambda, X} \le \tuple {\lambda', U}$, we have:

$\map \phi {\lambda, X} = \lambda \preceq \lambda' = \map \phi {\lambda', U}$

We show that $\family {x_{\map \phi m} }_{m \in \Omega}$ converges to $y$.

Let $U_0$ be an open neighborhood of $y$.

Take $\lambda_0 \in \Lambda$ such that $x_{\lambda_0} \in U_0$.

Then $\tuple {\lambda_0, U_0} \in \Omega$.

Take $\tuple {\lambda, U} \in \Omega$ such that $\tuple {\lambda_0, U_0} \le \tuple {\lambda, U}$.

Then $U_0 \supseteq U$ and $x_\lambda \in U$.

In particular, $x_{\map \phi {\lambda, U} } \in U \subseteq U_0$ whenever $\tuple {\lambda_0, U_0} \le \tuple {\lambda, U}$.

Since $U_0$ was an arbitrary open neighborhood of $y$, we have that $\family {x_{\map \phi m} }_{m \in \Omega}$ converges to $y$.

$\Box$

Sufficient Condition

Suppose that:

there exists:

a directed set $\struct {\Omega, \le}$

a monotone cofinal mapping $\phi : \Omega \to \Lambda$

such that the Moore-Smith sequence $\family {x_{\map \phi m} }_{m \in \Omega}$ converging to $y$.

Let $U$ be an open neighborhood of $y$.

Then there exists $m_U \in \Omega$ such that for all $m \in \Omega$ with $m_U \le m$, we have that $x_{\map \phi m} \in U$.

Now fix an open neighborhood $U$ of $y$ and $\lambda_0 \in \Lambda$.

Since $\phi$ is a cofinal mapping, there exists $u_0 \in \Omega$ such that $\lambda_0 \preceq \map \phi {u_0}$.

Since $\struct {\Omega, \le}$ is a directed set, there exists $u^\ast \in \Omega$ with $u_U \le u^\ast$ and $u_0 \le u^\ast$.

Since $\phi$ is monotone, we have $\lambda_0 \preceq \map \phi {u_0} \preceq \map \phi {u^\ast}$.

Since we also have $u_U \le u^\ast$, we have that $x_{\map \phi {u^\ast} } \in U$.

Setting $\lambda^\ast \in \Lambda$ we have $\lambda_0 \preceq \lambda^\ast$ such that $x_{\lambda^\ast} \in U$.

So $y$ is a cluster point of $\family {x_\lambda}_{\lambda \in \Lambda}$.