Question

Let $X$ be an uncountable set with the discrete topology, or the one-point compactification of such a set. Then ${\mathcal{B}}_X\ne {\mathbb{B}}_X$.

Short answer

For both discrete $X$ and its one-point compactification, ${\mathcal{B}}_X\ne {\mathbb{B}}_X$ due to the presence of different clopen sets.

Step-by-step solution

Understanding the Problem

For an uncountable set $X$ with the discrete topology, every subset is an open set. In this topology, the Borel $\sigma$-algebra ${\mathcal{B}}_X$ is generated by the open sets of $X$. The smallest $\sigma$-algebra containing all the open sets is simply the power set of $X$, because $X$ itself is uncountable and any subset of it can be taken as an open set.

Defining \sigma-Algebra Terms

There are two $\sigma$-algebras to consider: ${\mathcal{B}}_X$, the Borel $\sigma$-algebra, and ${\mathbb{B}}_X$, the smallest $\sigma$-algebra containing all Baire sets. In the discrete topology case for an uncountable $X$, ${\mathcal{B}}_X=\mathcal{P}(X)$, the power set of $X$.

Exploring The One-Point Compactification

The one-point compactification $Y=X\cup \{\mathrm{\infty }\}$ involves $Y$ having all subsets of $X$ plus sets of the form $X^c\cup \{\mathrm{\infty }\}$ as open sets. ${\mathcal{B}}_Y$, the Borel $\sigma$-algebra in this case, also contains all subsets of $X$ and open neighborhoods of $\mathrm{\infty }$.

Comparing \sigma-Algebras

For the discrete case, ${\mathcal{B}}_X$ is the power set of $X$, uncountably infinite. For the one-point compactification $Y$, the $\sigma$-algebra ${\mathbb{B}}_Y$ is indeed uncountably infinite but has a different structure from ${\mathcal{B}}_X$ due to the neighborhood basis at $\mathrm{\infty }$. This means ${\mathbb{B}}_Y$ includes certain clopen sets not in ${\mathcal{B}}_X$, leading to ${\mathcal{B}}_{Y}eq{\mathbb{B}}_Y$.

Conclusion

Hence, whether considering the original set or its one-point compactification, the Borel $\sigma$-algebra is not equivalent to the Baire $\sigma$-algebra due to the different types of infinity and way the neighborhoods at $\mathrm{\infty }$ are handled.

Key concepts

Discrete Topology

In discrete topology, every subset of a given set is considered an open set. When applied to an uncountable set, such as $X$ in this context, the concept becomes more intriguing.
This is because the power set of $X$, which contains all possible subsets, becomes equivalent to the Borel $\sigma$-algebra, ${\mathcal{B}}_X$.

• **Open Sets**: In discrete topology, every subset of $X$ is open. Given $X$'s uncountable nature, it means ${\mathcal{B}}_X$ = $\mathcal{P}(X)$.


• **Borel $\sigma$-Algebra**: Since every subset is open, the Borel $\sigma$-algebra is the power set itself.


• **Power Set**: The totality of every possible subset, even if $X$ is large like an uncountable set.

This is key in explaining why the structures of ${\mathcal{B}}_X$ and ${\mathbb{B}}_X$ differ when the topology is changed to its one-point compactification.

One-Point Compactification

One-point compactification is a method of adding a single 'point at infinity' to an otherwise non-compact space, such as $X$.
It is often denoted as $Y=X\cup \{\mathrm{\infty }\}$, transforming the topology remarkably.

• **Structure**: $Y$ includes all subsets of $X$ and new sets of the form $X^c\cup \{\mathrm{\infty }\}$. By doing this, $Y$ effectively becomes compact.


• **Open Sets**: Besides the subsets from $X$, which remain open, we introduce the neighborhoods of this new point $\mathrm{\infty }$.


• **Impact on $\sigma$-Algebra**: The Borel $\sigma$-algebra ${\mathcal{B}}_Y$ now includes these neighborhoods, resulting in a richer algebra compared to the original discrete case.

This demonstrates the alteration in structure and leads to differences in comparison to the purely discrete topology version of $X$.

Baire Sigma-Algebra

Baire $\sigma$-algebra often differs significantly from Borel $\sigma$-algebra, especially in contexts such as the ones described here.
While ${\mathcal{B}}_X$ in the discrete scenario was straightforward, the one-point compactification brings complexity.

• **Baire Sets**: Constructed through operations on basic open sets (like countable unions and intersections), they lead to a different structure in compact spaces.


• **Comparison with Borel $\sigma$-Algebra**: In $Y$, ${\mathbb{B}}_Y$ encompasses different sets including those open neighborhoods of $\mathrm{\infty }$. These often mix open and closed elements, termed "clopen" sets.


• **Implication**: This setup leads to the result ${\mathcal{B}}_{Y}eq{\mathbb{B}}_Y$ as some sets belong uniquely to one and not the other.

Baire $\sigma$-algebra’s contrasting behavior reflects how compactification changes the nature of the setting, differentiating it even from the vast power set of $X$ in discrete topology.