Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

In one sentence

Five spaces refute most of what can plausibly be claimed in general topology, and knowing which property each one breaks is faster than reasoning from definitions.

Why the exam asks it

Because a general-topology option is almost always false, and the efficient way to answer is to recognise which standard space kills it. This chapter is a lookup table with the reasoning attached.

The idea

Cofinite topology on an infinite set

Open sets: $\emptyset$ and the complements of finite sets.

• $T_{1}$: singletons are closed, since their complements are cofinite.
• Not Hausdorff: any two non-empty open sets have finite complements, so they must meet.
• Compact: given a cover, one member misses only finitely many points, and finitely many more members cover those.
• Every subset is compact, by the same argument. So compact subsets are very far from closed.
• Connected, since no two non-empty open sets are disjoint.

This is the workhorse for anything of the form "compact $\Rightarrow$ closed" or "$T_{1}$ $\Rightarrow$ $T_{2}$".

Cocountable topology on an uncountable set

Open sets: $\emptyset$ and complements of countable sets.

• $T_{1}$, not Hausdorff, connected, for the same reasons.
• Not first countable, and convergent sequences are eventually constant — so sequences detect nothing. This is the space for "sequences do not determine the topology".
• Not compact on $\mathbb{R}$: cover by complements of finite subsets of a countable set.

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

Basis: half-open intervals $[a,b)$.

• Finer than the usual topology, since $(a,b) = \bigcup [a+\tfrac1n, b)$.
• First countable, separable ($\mathbb{Q}$ is dense), Lindelöf, normal, totally disconnected.
• Not second countable: any basis needs a set with least element $a$ for each real $a$.
• Not metrisable: by Urysohn, regular plus second countable would give metrisability, and a separable metric space is second countable. Failing that, it cannot be metrisable.
• Not compact, and not locally compact.

The Sorgenfrey line is the standard example of separable-but-not-second-countable, and of a perfectly normal space with no metric.

The Sorgenfrey plane $\mathbb{R}_\ell^{2}$

• Separable, being a product of separable spaces.
• Not normal: the antidiagonal $\{(x,-x)\}$ is an uncountable closed discrete subspace, and a separable normal space cannot contain one of size continuum — Jones's lemma.
• Not Lindelöf.

So: the product of two normal spaces need not be normal, and the product of two Lindelöf spaces need not be Lindelöf. That single row refutes two plausible statements at once, which is why it is worth carrying.

The Cantor set

Remove middle thirds from $[0,1]$ forever. What is removed totals $\sum 2^{n-1}/3^n = 1$, so the remainder has measure zero.

• Compact (closed and bounded), totally disconnected, perfect (no isolated points), metrisable.
• Uncountable, of cardinality $2^{\aleph_0}$ — its points are exactly the numbers with a base-$3$ expansion using only $0$ and $2$.
• Homeomorphic to $\{0,1\}^{\omega}$, and by Tychonoff that gives compactness a second way.
• Characterisation: every non-empty compact, perfect, totally disconnected, metrisable space is homeomorphic to the Cantor set. Rare and worth knowing, since it makes "is this the Cantor set" a checkable question.

The fat Cantor set removes intervals of length $4^{-n}$ instead, totalling $\tfrac12$. It is still nowhere dense and still contains no interval, and it has measure $\tfrac12$ — so nowhere dense says nothing about measure.

Where intuition breaks

"$T_{1}$ implies Hausdorff." Cofinite, cocountable.

"Compact implies closed." Cofinite: every subset is compact.

"Separable implies second countable." Sorgenfrey line. True for metric spaces, which is where the instinct comes from.

"Normality is preserved by products." Sorgenfrey plane.

"A countable space is not compact" or "a compact space is not countable." Neither follows. Cofinite on a countable set is countable and compact.

"Measure zero implies countable." The Cantor set is uncountable with measure zero.

"Nowhere dense implies measure zero." The fat Cantor set has measure $\tfrac12$.

"A perfect set is an interval." The Cantor set is perfect and totally disconnected.

"Sequences determine closure." Cocountable on $\mathbb{R}$: convergent sequences are eventually constant, so sequential closure is trivial while topological closure is not.

The exam's angle

1. Read the option and ask which space kills it. Faster than reasoning from the definitions, and the list is short.
2. $T_{1}$ but not $T_{2}$ $\to$ cofinite. Separable but not second countable $\to$ Sorgenfrey. Product breaking a property $\to$ Sorgenfrey plane.
3. Anything about smallness $\to$ Cantor, fat Cantor, $\mathbb{Q}$. Countable, measure zero and nowhere dense are three independent notions.
4. Anything about sequences $\to$ cocountable, where they are useless.
5. Check whether the option secretly assumes a metric. Most of these are true in metric spaces, which is exactly why they look right.

The night before

• Cofinite (infinite set$): T_{1}$, not $T_{2}$, compact, every subset compact, connected.
• Cocountable (uncountable set$): T_{1}$, not $T_{2}$, connected, not first countable, convergent sequences eventually constant.
• Sorgenfrey line: basis $[a,b)$; separable, Lindelöf, normal, first countable, totally disconnected; not second countable, not metrisable, not compact.
• Sorgenfrey plane: separable, not normal, not Lindelöf. Products preserve neither.
• Cantor set: compact, perfect, totally disconnected, metrisable, uncountable, measure zero, $\cong\{0,1\}^\omega$. Characterised by those four properties.
• Fat Cantor set: nowhere dense with measure $\tfrac12$.
• Countable, measure zero, nowhere dense: three independent kinds of small.