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Let $X = \prod X_\lambda$ be a product of compact topological spaces. I want to show that $X$ is compact using nets. For that let $(x_\alpha)$ be a net in $X$.

Each net $(\pi_\lambda(x_\alpha))_{\alpha \in A}$ has a convergent subnet $(z_{\lambda, \beta})_{\beta \in B_\lambda}$, so that there is, for each $\lambda$ a monotone final function $f_\lambda : B_\lambda \to A$ where each $B_\lambda$ is a directed set.

I want to build out of this a convergent subnet of $(x_\alpha)$ but I'm having trouble with this because each subnet is defined on a different directed set.

In the finite product case this is simple. Suppose $X = X_1\times X_2$. Then a net $(x_\alpha)$ is

$$(x_{1,\alpha},x_{2,\alpha})_{\alpha \in A}$$

Thus we first look at $(x_{1,\alpha})$. Since $X_1$ is compact, this net has a convergent subnet $(z_{1,\beta})_{\in B_1}$ where there's $f_1 : B_1\to A$ monotone and final with $z_{1,\beta}=x_{1,f_1(\beta)}$. Thus we have the subet of $(x_\alpha)$

$$(x_{1,f_1(\beta)},x_{2,f_1(\beta)})_{\beta \in B_1},$$

Now, since $X_2$ is compact $(x_{2,f_1(\beta)})_{\beta \in B_1}$ has a convergent subnet $(z_{2,\beta})_{\beta \in B_2}$, where there's $f_2 : B_2\to B_1$ monotone and final with $z_{2,\beta}=x_{2,f_{2}\circ f_1(\beta)}$.

Since $(x_{1,f_1(\beta)})_{\beta \in B_1}$ converges every subnet converges and $(x_{1,f_2\circ f_1(\beta)})$ converges. This leads to

$$(x_{1,f_2\circ f_1(\beta)}, x_{2,f_2\circ f_1(\beta)})_{\beta \in B_2}$$

a convergent subnet of $(x_{1,\alpha},x_{2,\alpha})_{\alpha \in A}$.

The point is, on the finite case we can say "first deal with $X_1$, find a subnet, and with it produce a subnet on $X$, then deal with $X_2$, find another subnet and with it produce a new subnet on $X$, and so forth".

Indeed, this also tackles the countable case.

In the general case there's no meaning to say "first pick $\lambda_1$, then $\lambda_2$", because $\Lambda$ might not be a countable index set.

So how can I transfer this idea to the general case, and out of all subnets, with directed sets $B_\lambda$, produce a subnet on $X$?

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HINT: Imitate the construction of the product topology on $X$. For each $\lambda\in\Lambda$ fix a $c_\lambda\in B_\lambda$, let

$$B=\left\{b\in\prod_{\lambda\in\Lambda}B_\lambda:\{\lambda\in\Lambda:b_\lambda\ne c_\lambda\}\text{ is finite}\right\}\;;$$

the natural product order that $B$ inherits from $\prod_{\lambda\in\Lambda}B_\lambda$ makes it a directed set. For each $b\in B$ let $\operatorname{supp}(b)=\{\lambda\in\Lambda:b_\lambda\ne c_\lambda\}$; there is an $a_b\in A$ such that $f_\lambda(b_\lambda)\le a_b$ for each $\lambda\in\operatorname{supp}(b)$. Now define $f:B\to A:b\mapsto a_b$.

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