Problem in proof of -"A net has $y$ as a cluster point iff it has a subnet which converges to $y$"

Question

Directed Set:

We say that $(\omega, ≤)$ is a directed set, if ≤ is a relation on $\omega$ such that

(i) x ≤ y ∧ y ≤ z ⇒ x ≤ z for each x, y, z ∈ $\omega$;

(ii) x ≤ x for each x ∈ $\omega$;

(iii) for each x, y ∈ $\omega$ there exist z ∈ $\omega$ with x ≤ z and y ≤ z

A net has $y$ as a cluster point iff it has a subnet which converges to $y$

Proof:Let y be a cluster point of $(x_{\lambda})$.

Define, M={$(\lambda,U):\lambda \in \omega$,U is a nbhd of $y$ such that $x_{\lambda}\in U$} and order M as follows $(\lambda_1,U_1)\le (\lambda_2,U_2)$ iff $\lambda_1 \le\lambda_2$ and $U_2\subset U_1$.

Now we will show that M is a directed set.

Reflexivity:

Let $(\lambda,U)\in M$ then $\lambda\in \omega$(directed set ) and U is a nbhd of $y$ s.t. $x_{\lambda}\in U$,then $\lambda\le \lambda$ and $U\subset U$

($U_1=U_2\leftrightarrow U_1\subset U_2,U_2\subset U_1$ )

$(\lambda,U)\le (\lambda,U)$

Transitivity:

Let $(\lambda_1,U_1),(\lambda_2,U_2),(\lambda_3,U_3)\in M$ s.t.

$(\lambda_1,U_1)\le (\lambda_2,U_2)$ and $(\lambda_2,U_2)\le (\lambda_3,U_3)\implies \lambda_1\le \lambda_2,U_2\subset U_1;\lambda_2\le \lambda_3,U_3\subset U_2$.

Since,$\lambda_1,\lambda_2,\lambda_3\in \omega$.So, $\lambda_1\le\lambda_2,\lambda_2\le\lambda_3\implies\lambda_1\le \lambda_3$

and by the transitivity of subset relation ,we have $U_2\subset U_1;U_3\subset U_2$,so $U_3\subset U_ 2\subset U_1\implies U_3\subset U_1$.So,$\lambda_1\le \lambda_3; U_3\subset U_1\implies (\lambda_1,U_1)\le (\lambda_3,U_3)$

Property(iii):

if $(\lambda_1,U_1),(\lambda_2,U_2)\in M$ ,then By the first property of directed set $\omega,\lambda_1,\lambda_2\in \omega \implies$ there exists $\lambda_3$ s.t. $\lambda_1\le \lambda_3$ and $\lambda_2\le \lambda_3$ and by the property of nbhd of $y$ there exists $U_3\subset U_1$ and $U_3\subset U_2$.

I know in order to show $x_{{\lambda}_3} \in U_3$ we have to use the assumption that "y is a cluster point of net",but i'm not getting how to write this fact mathematically?

Answer 1

Let $(\lambda_{1},U_{1})$, $(\lambda_{2},U_{2})\in M$. Take $U_{3}=U_{1}\cap U_{2}$, then $U_{3}$ is a neighborhood of $y$. By the third property of a directed system, there exists $\lambda'\in\omega$ such that $\lambda_{1}\leq\lambda'$ and $\lambda_{2}\leq\lambda'$. Since $y$ is a cluster point, for the neighborhood $U_{3}$ and $\lambda'$, there exists $\lambda_{3}\in\omega$ such that $\lambda'\leq\lambda_{3}$ and $x_{\lambda_{3}}\in U_{3}.$ Then $(\lambda_{3},U_{3})$ is an element in $M$ that satisfies $(\lambda_{1},U_{1})\leq(\lambda_{3},U_{3})$ and $(\lambda_{2},U_{2})\leq(\lambda_{3},U_{3})$.