# What are the components of the Stone-Cech Remainder?

Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a discrete topological space and consider the continuous map $F:X_i \to I$ that sends each $X_i$ to $i \in I$. The Stone Cech lift $\beta F: \beta X \to \beta I$ restricts to a surjection $G: X^* \to I^*$. For each free ultrafilter $\mathcal U \in I^*$ the preimage $G^{-1}(\mathcal U)$ is a subcontinuum of $X^*$ and $\{G^{-1}(\mathcal U): \mathcal U \in I^* \}$ is the set of components. For a proof see Lemma 2.1.

This fact is used to prove each subcontinuum of the remainder $\mathbb H ^*$ of the half line is the intersection of all the standard subcontinua containing it (Theorem 5.1).

Definition: Suppose $Y$ is a locally compact noncompact Tychonoff space. By a standard subcontinuum we mean a set of the form $G^{-1}(\mathcal U)$ where $X_i$ are subcontinua of $Y$ and $\bigcup X_i$ has the discrete union topology.

Suppose $K \subset \mathbb H$ is a subcontinuum. For each $x \in \mathbb H ^* - x$ let $U^*$ be open at $x$ and disjoint from $K$. It follows $\mathbb H - U^*$ is the disjoint union of infinitely many intervals with the discrete union topology. By the above the components of $\mathbb H - U^*$ are all standard subcontinua and one of them contains $K$ so is disjoint from $U$.

Suppose $Y$ is a more general space than a discrete union of intervals. $Y$ is locally-compact, noncompact, Hausdorff and each of the infinitely many components is compact. Suppose we take a collection $\{X_i: i \in I \}$ of components and an ultrafilter $\mathcal U \in I^*$ with the following property: For each compact $K \subset Y$ there is $U \in \mathcal U$ with $\bigcup \{X_i: i \in U\}$ disjoint from $K$. (observe this holds for a disjoint union of intervals simultaneously for all $\mathcal U \in I^*$) Then the set $G^{-1}(\mathcal U)$ is a subcontinuum of $Y^*$ but I see no reason it should be a component.

Is there a more general version of Lemma 2.1 linked above? More generally what is known about expressing the components of Stone-Cech remainders in terms of the components of the original space? Is there any reason to believe a version of Theorem 5.1 should hold for more general locally compact noncompact connected Hausdorff spaces than $\mathbb H$?

Edit: Is there any known theorem something like this?

Conjecture: Suppose $Y$ is a Tychonoff space and each component is compact. The quasicomponents of $\beta Y$ correspond to the ultrafilters of clopen sets of $Y$.

The quasicomponent of a point means the intersection of all clopen sets at that point.

• Regarding your conjecture: (1) In $\beta Y$ quasicomponents and components are the same thing. (2) The quasicomponent decomposition space $Y/\sim$ is zero dimensional Tychonoff. Let $\beta\varphi:\beta Y\to \beta(Y/\sim)$ extend the epimorphism $\varphi:Y\to Y/\sim$. Then I think (?) what you are asking is: Are the components of $\beta Y$ the same as the non-empty point inverses of $\beta\varphi$? (3) I wonder how this goes if $Y$ is zero-dimensional but not strongly zero dimensional? Here $\beta Y$ contains a non-degenerate continuum, though $Y$ has very fine clopen structure. – D.S. Lipham May 7 '18 at 4:12

Let $Y$ be Tychonoff and $p\in \beta Y$.
Using the facts that (1) $A=\overline{A\cap Y}$ for every clopen $A\subseteq \beta Y$, and (2) every clopen subset of $Y$ is a zero set, we can see that
$$\{A\subseteq \beta Y:A\text{ is clopen and }p\in A\}=\{\overline {A\cap Y}:A\subseteq \beta Y\text{ is clopen and }A\cap Y\in p\}=\{\overline B:B\subseteq Y \text{ is clopen and }B\in p\}.$$
This shows the one-to-one correspondence between quasi-components of $\beta Y$ ($=$ components of $\beta Y$) and ultrafilters of clopen subsets of $Y$. Namely, the quasi-component of $p$ is equal to $$\bigcap\{\overline B:B\in u\},\text{ where } u=\{B\subseteq Y:B\text{ is clopen and }B\in p\}$$ is an ultrafilter of clopen subsets of $Y$.