Sorgenfrey Lower Limit Topology

• Sorgenfrey lower limit topology is defined on ℝ using basis elements of the form [x, x+ε) and is recognized for its atypical, pathological properties compared to the Euclidean topology.
• It exhibits key characteristics such as zero-dimensionality, first-countability, and hereditary Lindelöfness, while lacking local compactness and metrizability.
• In product spaces, the Sorgenfrey line demonstrates complex behavior with failure of normality, underscoring deep connections between combinatorial set theory and topological separation axioms.

The Sorgenfrey lower limit topology, denoted $(\mathbb{R}, \tau_S)$ or $\mathbb{R}[\leq]$, is the topology on $\mathbb{R}$ generated by the basis of all half-open intervals of the form $[x, x+\epsilon)$ for $x \in \mathbb{R}$ and $\epsilon > 0$. This topology is finer than the standard Euclidean topology and serves as a canonical example illustrating a range of pathological and distinctive behaviors in general topology, particularly in relation to Lindelöfness, separation properties, metrizability, and normality in topological products.

1. Definition and Basic Properties

Given a set $X \subseteq \mathbb{R}$, the Sorgenfrey lower limit topology $X[\leq]$ is defined by the subbasis consisting of sets $[a,b) \cap X$ for $a < b$. For $X = \mathbb{R}$, this yields the Sorgenfrey line $(\mathbb{R},\tau_S)$, with the standard basis $$\mathcal{B}_S = \{ [x, x+\epsilon) : x \in \mathbb{R},\, \epsilon > 0 \}$$. Every $[x, x+\epsilon)$ is both open and closed (clopen) in this topology. The Sorgenfrey line is strictly finer than the Euclidean topology and is not locally compact, not $\sigma$-compact, non-metrizable, but it is zero-dimensional, first-countable, quasi-metrizable, hereditarily Lindelöf, and perfectly normal (Lin et al., 2018).

2. Separation Axioms and Zero-Dimensionality

$(\mathbb{R}, \tau_S)$ is zero-dimensional: it admits a basis of clopen sets as every basic half-open interval $[x, x+\epsilon)$ is clopen. More generally, for an $H$-space $(\mathbb{R},\tau_A)$ interpolating between the Euclidean and Sorgenfrey topologies, zero-dimensionality occurs if and only if $\mathbb{R}\setminus A$ is dense in the Euclidean topology. Specializing to $A = \emptyset$ yields the classical Sorgenfrey line, which is zero-dimensional because every point has a neighborhood base of clopen intervals (Lin et al., 2018).

Perfect normality holds for $(\mathbb{R},\tau_S)$, making it the prototypical example of a Lindelöf, non-metrizable, perfectly normal space. However, higher products exhibit a sharp failure of normality.

3. Compactness Properties

The Sorgenfrey line $(\mathbb{R},\tau_S)$ is neither locally compact nor a $k_\omega$-space, as the entire space is closed but not discrete (points accumulate from the right in any neighborhood basis). $(\mathbb{R},\tau_S)$ is not $\sigma$-compact: it cannot be written as a countable union of compact subspaces, because each compact in the Sorgenfrey topology must be countable and discrete. In general, for $H$-spaces $(\mathbb{R},\tau_A)$, $\sigma$-compactness requires $\mathbb{R}\setminus A$ to be countable and scattered, a condition not met for $A = \emptyset$ (Lin et al., 2018).

4. Metrizability, First-Countability, and Quasi-Metrizability

$(\mathbb{R},\tau_S)$ is first-countable; every $x$ admits a countable neighborhood basis $\{[x, x+1/n): n \in \mathbb{N} \}$. However, it is not metrizable, as metrizability would imply the existence of a countable base for the topology, which is precluded by the uncountable number of pairwise disjoint basic open sets. $(\mathbb{R},\tau_S)$ is not a $\beta$-space, as this is equivalent (in this framework) to metrizability or, for $H$-spaces, to $\mathbb{R}\setminus A$ being countable (Lin et al., 2018).

Despite non-metrizability, $(\mathbb{R},\tau_S)$ is quasi-metrizable: the quasi-metric $d(x, y) = \max\{0, y-x\}$ (Sinclair’s quasi-metric) defines the topology. In the general $H$-space framework, quasi-metrizability holds if $\mathbb{R}\setminus A$ is an $F_\sigma$ in the reverse Sorgenfrey topology.

5. Normality in Products and the Role of Special Sets

The square of the Sorgenfrey line $(\mathbb{R}[\leq])^2$ is the first major example of a hereditarily Lindelöf, separable, perfectly normal, but non-normal space. The lack of normality is a rank-2 pathology inherent to the product structure. However, for appropriate subsets $X \subseteq \mathbb{R}$, normality is restored in the square if and only if $X$ has strong combinatorial regularity properties (Szeptycki et al., 15 Nov 2025):

• Q-sets: If $X$ is a Q-set (every $A\subseteq X$ is a $G_\delta$ in $X$), then $(X[\leq])^2$ is normal. The separation argument relies on the fact that every closed discrete subset arises as the graph of a strictly decreasing function and can be separated via classical “shoelace” lemmas.
• $\lambda$-sets: If $X$ is a $\lambda$-set (every countable $A\subseteq X$ is $G_\delta$), then $(X[\leq])^2$ is pseudo-normal: for any two closed subsets, if at least one is countable, there exist disjoint open neighborhoods separating them. This generalizes properties from the Moore–Niemytzki plane.
• Independence phenomena: Under the Continuum Hypothesis (CH), it is possible to construct uncountable sets $X$ concentrated on a countable dense set so that $(X[\leq])^2$ is normal, even though $X$ is neither a Q-set nor a $\lambda$-set. This is achieved via entangled sets and combinatorial arguments extending classical Lusin–Sierpiński constructions.

The table summarizes product normality for various subsets:

Subset of $\mathbb{R}$	$(X[\leq])^2$ normal?	$(X[\leq])^2$ pseudo-normal?
Full $\mathbb{R}$	No	No
Q-set	Yes	Yes
$\lambda$-set	No	Yes
Special (CH) set	Yes	Varies

Normality in higher products is pathway-dependent and tightly connected to set-theoretic properties of $X$ (Szeptycki et al., 15 Nov 2025).

6. Subparacompactness and Countable Products

For every $n \in \mathbb{N}$ and any $A \subseteq \mathbb{R}$, the $n$-fold product $(\mathbb{R},\tau_A)^n$ is perfectly subparacompact (i.e., perfect and subparacompact). In particular, all countable products of the Sorgenfrey line are perfectly subparacompact, even though they are not normal. This property is preserved via induction on $n$, employing the hereditary Lindelöf property and refining covers known from the base case (Lin et al., 2018). This behavior distinguishes the Sorgenfrey product structure from that of metrizable or locally compact spaces.

7. Broader Significance and Pathological Features

The Sorgenfrey lower limit topology, and especially its products, illustrate the deep interplay between combinatorial descriptive set theory and topological separation axioms. The pathological properties—such as the failure of normality in products and the absence of $\sigma$-compactness and local compactness—underscore the limitations of intuition from metrizable spaces when extended to finer or non-metrizable topologies. Results connecting Q-sets, $\lambda$-sets, and entangled sets to product separation properties exemplify the bridge between set-theoretic topology and classical analysis. Strong forcing axioms, such as PFA and Baumgartner’s Axiom, further dictate the structure of non-normal products and the prevalence of large Q-sets (Szeptycki et al., 15 Nov 2025). These phenomena demonstrate the extent to which higher cardinal pathologies can be "localized" or eliminated by restricting attention to suitably regular subsets of $\mathbb{R}$.