Question

Let \(Q\) have the relative topology induced from \(R\). a. \(\mathbb{Q}\) is not locally compact. b. \(\mathbb{Q}\) is \(\sigma\)-compact (it is a countable union of singleton sets), but uniform convergence on singletons (i.e., pointwise convergence) does not imply uniform convergence on compact subsets of \(Q\).

Short answer

\( \mathbb{Q} \) is not locally compact but \( \sigma \)-compact. Uniform convergence on singletons doesn't imply uniform convergence on compact subsets.

Step-by-step solution

Understanding Q in Relative Topology

The set \( \mathbb{Q} \) refers to the set of all rational numbers, and it is given the relative topology induced from \( \mathbb{R} \). This means \( \mathbb{Q} \) inherits its topology from the standard topology on \( \mathbb{R} \) (related to open intervals).

Analyzing Local Compactness

To determine if \( \mathbb{Q} \) is locally compact, we must see if every point in \( \mathbb{Q} \) has a neighborhood basis made up of compact sets. A set is compact if it is closed and bounded (Heine-Borel Theorem for \( \mathbb{R} \)), however, no interval of rationals is closed in \( \mathbb{R} \). Therefore, \( \mathbb{Q} \) is not locally compact as such intervals can't be compact.

Checking \(\sigma\)-compactness

A space is \( \sigma \)-compact if it can be expressed as a countable union of compact subsets. Consider the union of singleton sets, where each singleton set contains one rational number. Since singleton sets are compact, \( \mathbb{Q} \) is \( \sigma \)-compact because \( \mathbb{Q} \) is a countable union of such sets.

Considering Uniform Convergence on Singletons

Uniform convergence implies that the rate of convergence of a sequence of functions does not depend on the choice of point within the compact set. For a sequence of functions \( f_n \to f \) pointwise and uniformly on singletons, the convergence does not necessarily imply uniform convergence on any compact subset of \( \mathbb{Q} \) because compact subsets of \( \mathbb{Q} \) would need to include more than one point.

Key concepts

Rational Numbers

Rational numbers, often denoted as \( \mathbb{Q} \), are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Symbolically, any rational number can be written as \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). This set of numbers encompasses values like \( \frac{1}{2}, \frac{-3}{4}, \) and \( 5 \).
Rational numbers are dense in the real numbers \( \mathbb{R} \), meaning between any two real numbers, there is at least one rational number. Topologically speaking, when \( \mathbb{Q} \) is given the relative topology induced from \( \mathbb{R} \), it inherits properties such as the non-compactness of open intervals, despite its countability. Understanding \( \mathbb{Q} \) with this relative topology allows us to explore interesting properties such as local compactness and \( \sigma \)-compactness.

Local Compactness

A topological space is said to be locally compact if every point has a base of neighborhoods that are compact. In simpler terms, around every point, there's a neighborhood that not only contains the point but is also compact. In the case of \( \mathbb{R} \), compactness is characterized by the Heine-Borel theorem which states that a subset is compact if it is closed and bounded.
However, for \( \mathbb{Q} \) (rational numbers), achieving local compactness poses difficulties. Although \( \mathbb{Q} \) is dense in \( \mathbb{R} \), no open interval fully composed of rationals is closed in \( \mathbb{R} \). This property hinders any neighborhood from being locally compact since compactness is unattainable. Therefore, \( \mathbb{Q} \) isn't locally compact because it cannot satisfy the compactness criteria innately built into \( \mathbb{R} \)'s topological structure.

Sigma Compactness

Sigma (\( \sigma \)) compactness is an intriguing property of a topological space. A space is considered \( \sigma \)-compact if it can be expressed as a countable union of compact sets. For \( \mathbb{Q} \), this condition is satisfied by seeing \( \mathbb{Q} \) as a union of singleton sets. Each singleton \( \{q\} \), where \( q \) is a rational number, is compact because a single point is trivially both closed and bounded.
Since \( \mathbb{Q} \) is countably infinite, it can be entirely covered by these compact singletons. Thus, \( \mathbb{Q} \) demonstrates \( \sigma \)-compactness by being a countable union of sets that are individually compact. This property highlights an essential aspect of rational numbers' topological structure: although \( \mathbb{Q} \) isn't locally compact, it can still be decomposed into compact components.

Uniform Convergence

Uniform convergence is a concept referring to a sequence of functions \( f_n \) that converges to a function \( f \) such that the speed of convergence is the same across the entire domain. In mathematical terms, for any small positive number \( \epsilon \), there exists a point in the sequence beyond which the absolute difference between \( f_n(x) \) and \( f(x) \) is less than \( \epsilon \) for all points \( x \) in a considered subset.
In the context of \( \mathbb{Q} \), uniform convergence is complicated by its lack of local compactness. While sequences may converge pointwise on singleton sets, achieving uniform convergence across any compact subset is not straightforward due to the non-existence of broader compact neighborhoods. Consequently, a sequence converging uniformly on singletons does not imply uniform convergence on larger, compact subsets. This distinction underscores the nuanced behavior of convergence in spaces with non-trivial topologies, such as \( \mathbb{Q} \).