Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

The property table

Property	Cofinite on $\mathbb{R}$	Cocountable on $\mathbb{R}$	Sorgenfrey $\mathbb{R}_\ell$	Cantor set
$T_{1}$	✓	✓	✓	✓
Hausdorff	✗	✗	✓	✓
compact	✓	✗	✗	✓
connected	✓	✓	✗	✗
separable	✓	✗	✓	✓
first countable	✗	✗	✓	✓
second countable	✗	✗	✗	✓
metrizable	✗	✗	✗	✓
normal	✓	✓	✓ (square is not)	✓

Cofinite topology

Open = ∅ or complement of a finite set. On an infinite set: $T_{1}$ but not Hausdorff, compact, connected, and every injective map into it is continuous. Convergence is strange: every sequence of distinct points converges to every point.

Sorgenfrey line $\mathbb{R}_\ell$

Base [a, b). Finer than the usual topology. Separable, first countable, Lindelöf, normal, totally disconnected — but not second countable and not metrizable. Its square is separable but contains an uncountable discrete subspace, so $\mathbb{R}_\ell^{2}$ is not normal: the standard counterexample to "product of normal is normal".

Cantor set

Compact, perfect (no isolated points), totally disconnected, uncountable, measure zero, nowhere dense, homeomorphic to {0,1}$^\mathbb{N}. **$Every** compact metric totally disconnected perfect space is homeomorphic to it.

Key takeaways

• Need "compact but not Hausdorff"? Cofinite. "Connected but not Hausdorff"? Cofinite.
• Need "normal but square not normal"? Sorgenfrey.
• Need "uncountable but measure zero"? Cantor.