# 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. Commented May 7, 2018 at 4:12

The answer to this question contains two locally compact zero-dimensional spaces whose Cech-Stone compactification is not zero-dimensional.That may put a limit on what can be said about components of the remainder vis-a-vis the components of the space itself.

The conjecture is true, but I'll leave it to others to address the other parts of your question (or maybe I'll edit this post later).

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 $Y$.