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The cofinite topology on an infinite set is $T_{1}$ but neither Hausdorff nor regular nor normal

Example

Let $X$ be an infinite set — that is, a set that is not finite (Finite, countably infinite, countable, uncountable) — and give it the cofinite topology $T_{c o f} = {\emptyset} \cup {U \subseteq X : X ∖ U is finite}$ (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), whose closed sets are $X$ together with the finite subsets of $X$ . Then:

$(X, T_{c o f})$ is $T_{1}$ ( $T_{0}$ (Kolmogorov) and $T_{1}$ (Frechet) spaces).
No two nonempty open sets are disjoint. Consequently the space is not Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not), not regular (Regular spaces and $T_{3}$ spaces, with the source disagreement over whether regularity includes $T_{1}$ stated explicitly) and not normal (Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

So the cofinite topology on an infinite set satisfies $T_{1}$ and $T_{0}$ and fails every axiom above them. It is the standard witness that $T_{1}$ is strictly weaker than the Hausdorff condition, and that is how it is used on the main page (FALSE: every $T_{1}$ space is Hausdorff); here it is pushed further, to show that $T_{1}$ implies neither of the two axioms that sit above $T_{2}$ either.

Facts & Assumptions

Given: An infinite set $X$ with the cofinite topology $T_{c o f}$ , and points $x, y, z \in X$ .

[A1]

$U \in T_{c o f}$ exactly when $U = \emptyset$ or $X ∖ U$ is finite; the closed sets are $X$ and the finite subsets of $X$ ; and a union of two finite sets is finite (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, facts (i) and (ii) of that item, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[L1]

A topology is $T_{1}$ exactly when it contains the cofinite topology on the same set (A space is $T_{1}$ if and only if every singleton is closed, if and only if every finite subset is closed, if and only if its topology contains the cofinite topology, clause (d), $T_{0}$ (Kolmogorov) and $T_{1}$ (Frechet) spaces).

[L2]

Hausdorff: distinct points have disjoint open neighbourhoods. Regular: a point and a closed set not containing it have disjoint open supersets. Normal: two disjoint closed sets have disjoint open supersets (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Regular spaces and $T_{3}$ spaces, with the source disagreement over whether regularity includes $T_{1}$ stated explicitly, Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

[L3]

A set with at most one element is finite, being equinumerous with $0$ or with $1$ ; so an infinite set has at least three distinct points, since a set with at most two elements is finite as a union of two sets each with at most one (Finite, countably infinite, countable, uncountable, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, fact (ii)).

[L4]

A set is closed exactly when its complement is open (Interior, closure, boundary, exterior, derived set and isolated point in a topological space, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

Verification

technique · direct

1.1

$T_{c o f} \subseteq T_{c o f}$ , so $(X, T_{c o f})$ is $T_{1}$ , which is claim 1.

1.2

$X$ contains three distinct points $x$ , $y$ , $z$ .

1.3

Let $U, V$ be nonempty open sets and suppose $U \cap V = \emptyset$ ; then $X = (X ∖ U) \cup (X ∖ V)$ is a union of two finite sets, hence finite by [A1], contradicting the hypothesis that $X$ is infinite. So no two nonempty open sets are disjoint, which is the first half of claim 2.

A1assume-hyp

2.1

$x \neq y$ , and any open $U ∋ x$ and open $V ∋ y$ are nonempty, hence meet by step 1.3; so the space is not Hausdorff by [L2].

step 1.2step 1.3L2

2.2

${y}$ is closed by [A1] and $x \notin {y}$ ; any open $U ∋ x$ and open $V \supseteq {y}$ are nonempty, hence meet by step 1.3; so the space is not regular by [L2].

step 1.2step 1.3A1L2L4

2.3

${y}$ and ${z}$ are disjoint nonempty closed sets by [A1] and step 1.2; any open $U \supseteq {y}$ and open $V \supseteq {z}$ are nonempty, hence meet by step 1.3; so the space is not normal by [L2].

step 1.2step 1.3A1L2L4

3.1

Steps 2.1, 2.2 and 2.3 complete claim 2, and step 1.1 is claim 1.

step 1.1step 2.1step 2.2step 2.3∎

Remarks

The three failures have one cause. In the cofinite topology on an infinite set the open sets are so large that no two nonempty ones are disjoint, so every separation axiom whose conclusion is a pair of disjoint nonempty open sets fails at once. What survives is $T_{1}$ , whose conclusion asks for open sets that are allowed to overlap.
The hypothesis that $X$ is infinite is necessary. On a finite set the cofinite topology is the discrete one (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), which satisfies every axiom on the main page.
This is the coarsest $T_{1}$ topology on $X$ (A space is $T_{1}$ if and only if every singleton is closed, if and only if every finite subset is closed, if and only if its topology contains the cofinite topology, clause (d)), which is why it is the natural place to look for a $T_{1}$ space that fails everything else: any $T_{1}$ topology on $X$ contains it, so any counterexample to a $T_{1}$ implication should be sought here first.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cofiniteness (Wikipedia) (standard reference, not scraped)
Separation axiom (Wikipedia) (standard reference, not scraped)
L. Steen and J. Seebach, Counterexamples in Topology, §18 (standard reference, not scraped)
T1 space (Wikipedia) (standard reference, not scraped)

The cofinite topology on an infinite set is T1T1​ but neither Hausdorff nor regular nor normal