My understanding of the Stone-Cech Compactification of a space $X$ is as follows:
Let $\mathcal{F} = \{ f:X \to I_f \}$ so that $f$ is continuous and $I_f \subset \mathbb{R}$ is a compact interval. Define
$$e: X \to \prod_{f \in \mathcal{F}} I_f \hspace{0.2cm} \mathrm{by} \hspace{0.2cm} e(x) = \prod_{f \in \mathcal{F}} f(x)$$
So then
$$e^\to(X) = \Big\{ \displaystyle\prod_{f \in \mathcal{F}} f(x): x \in X \Big\}$$
Now, let $\overline{e^\to(X)} = \beta(X)$.
I've seen posts such as Stone-Cech Compactification and Connectedness say that $X \subset \beta(X)$ and that $\overline{X} = \beta(X)$.
But this isn't clear to me. So I am wondering:
- How is $X \subset \beta(X)$?
- How is $\overline{X} = \beta(X)$?
Using the definition I've been given above, it is hard to see how objects in $X$ are comparable to objects in $\beta(X)$.
You “identify” $X$ as a subspace of $\beta X$ via the map $e$. I put “identify” in quotes because $e$ is an embedding iff. $X$ is a Tychonoff space; $e$ is injective iff. $X$ is completely Hausdorff. But when operating on $X\subset\beta X$, you really mean you are working with the image $e(X)$.
Then $\overline{X}=\beta X$ is just standing for the statement that $e(X)$ is dense in $\beta X$, which is true by your definition.
To reiterate, if you write $x\in\beta X$ you “really mean” $e(x)\in\beta X$.