# Do Hausdorff spaces that aren't completely regular appear in practice?

Completely regular spaces include all metrizable spaces, topological vector spaces, and topological groups in general. In fact, they are exactly the uniformizable spaces. Complete regularity is hereditary, ie. a subspace of a completely regular space is also completely regular, and it's preserved by arbitrary products. The completely regular spaces are in fact a reflective subcategory of $\mathrm{Top}$, as can be seen from another characterization: they are exactly the spaces for which the real functions determine the topology, ie. the topology is the initial topology for the set of real continuous functions. Finally, every subspace of a compact Hausdorff space is completely regular, and conversely, every ($T_0$) completely regular space embeds universally into a compact Hausdorff space via the Stone-Čech compactification.

There are of course many natural examples of topologies that aren't completely regular. The two that I know of are Zariski topology, which is $T_1$, but not Hausdorff, and Alexandrov topology, which is a natural topology on a poset that can't even be $T_1$ in an interesting way.

What I haven't seen before are examples of Hausdorff topologies that aren't completely regular, and weren't constructed specifically for the purpose of being a counterexample. Considering how widely topology is applied (and how little I know of it) I'm assuming there are some, and I'd be interested in hearing how often they appear.

The strengthening of this question to normal spaces seems to have elementary and satisfactory answers: the topology of pointwise convergence on $\mathbb R$ isn't normal (witnessing the fact that normal spaces aren't closed under products).

• What does "in practice" mean? A functional analyst, an algebro-geometer, an algebro-topologist, a probabilist... will all have different notions of what "in practice" means. – Najib Idrissi Jun 9 '16 at 15:32
• @NajibIdrissi It's not a strictly defined term, but I don't think it's that vague. If any one these people encounters a topological space in their everyday work, I'd certainly count that topological space as appearing in practice. – user54748 Jun 9 '16 at 15:46
• It seems unlikely. Complete regularity is inherited by all of the simple operations: subspaces, products, inverse limits, etc. So a non-completely regular space has to be quite unrelated to most of the basic spaces that appear in practice. – Forever Mozart Jun 9 '16 at 20:59
• The spaces I have seen in functional analysis have nice properties. They are locally convex. Usually there is some relatively natural uniformity and uniformizable spaces are completely regular. – Martin Sleziak Feb 12 at 15:06

“Thirteen twenty-seven, at the beginning
of the first hour, on the sixth day of April,
I entered the labyrinth, and see no escape”.
Petrarch, ‘The Canzoniere’, Poem 211.

It's not a strictly defined term, but I don't think it's that vague. If any one these people encounters a topological space in their everyday work, I'd certainly count that topological space as appearing in practice.

Yes, I do this for about twenty years. I am a general topologist. Moreover, I deal with paratopological groups. I was directed to this topic by Igor Guran a supervisor of my PhD and now I’m finishing my habilitation thesis about them.

I recall the basis notions. Let $$G$$ be a group endowed with a topology $$\tau$$. A pair $$(G,\tau)$$ is called a semitopological group provided the multiplication $$\cdot:G\times G\to G$$ is separately continuous. Moreover, if the multiplication is continuous then $$(G,\tau)$$ is called a paratopological group. Moreover, if the inversion $$(\cdot)^{-1}:G\to G$$ is continuous with respect to the topology $$\tau$$, then $$(G,\tau)$$ is a topological group. A classical example of a paratopological group failing to be a topological group is the Sorgenfrey line, that is the real line endowed with the Sorgenfrey topology (generated by the base consisting of half-intervals $$[a,b)$$, $$a). Basic properties of semitopological and paratopological groups are described in [ArhTka], in my PhD thesis [Rav3] and papers [Rav] and [Rav2]. New survey [Tka2] presents recent advances in this area.

It is well known that almost all usual separation axioms are equivalent for a topological group, namely, each $$T_0$$ topological group is $$T_{3\frac 12}$$ and $$T_1$$. This non-trivial fact was established by Pontrjagin near 1936. On the other hand, simple examples (see, for instance, Examples 1.6-1.8 from [Rav]) show that for paratopological groups neither of the implications $$T_0\Rightarrow T_1 \Rightarrow T_2 \Rightarrow T_3$$ holds and only a few backwards implications between different separation axioms can be made, see, for instance [Rav2, Section 1] or [Tka2, Section 2], and a more general paper [Tka3]. In particular, classical constructions can be used to show that each $$T_3$$ paratopological group is $$T_{3\frac 12}$$ and each $$T_2$$ paratopological group is functionally $$T_2$$ [BanRav2]. We recall that a topological space $$X$$ is functionally $$T_2$$ if for any distinct points $$x,y\in X$$ there exists a continuous function $$f:X\to\Bbb R$$ such that $$f(x)\ne f(y)$$. Since used constructions are similar to that in Pontrjagin's proof, if these results were proved in 1937 then they would were trivial remarks. But it happened in 2014, and the first of these results solved, as far as I know, the oldest and most known problem in the theory of paratopological groups. For instance, discussing it in Section 2 of [Tka2] Tkachenko wrote that it is “open for about 60 years” and “the authority of the question is unknown, even if all specialists in the area have it in mind”. Moreover, this solution even earned two upvotes at MSE!

An important issue of separation axioms for the theory of paratopological groups is their affect on claims on automatic continuity of the inversion. This is one of main directions of the theory, and, as far as I know, the firstly developed that. It turned out that if a space of paratopological group satisfies some conditions (sometimes with some conditions imposed on the group) then the inversion in the group is continuous, that is the group is topological. An interested reader can find known results on this subject in Introduction of [AlaSan], in the survey Section 5.1 of [Rav3], in Section 3 of the survey [Tka2], and in [Rav4].

Thus, in 1936 Montgomery [Mon] showed that every completely metrizable paratopological group is a topological group. In 1953 Wallace [Wal] asked whether each locally compact regular semitopological group a topological group. In 1957 Ellis obtained a positive answer of the Wallace question (see [Ell1], [Ell2]). In 1960 Zelazko used Montgomery's result and showed that each completely metrizable semitopological group is a topological group. Since both regular locally compact and completely metrizable topological spaces are Čech-complete, this suggested Pfister [Pfi] in 1985 to ask whether each Čech-complete semitopological group is a topological group. In 1996 Bouziad [Bou] and Reznichenko [Rez2], as far as I know, independently obtained affirmative answers to the Pfister's question. To do this, it was sufficient to show that each Čech-complete semitopological group is a paratopological group since earlier, Brand [Bra] had proved that every Čech-complete paratopological group is a topological group. Brand's proof was later improved and simplified in [Pfi].

We can see here the role of separation axioms as follows. Each (not necessarily) regular locally compact paratopological group is a topological group, see Proposition 5.5 in [Rav3] or its counterpart in English in this answer.

Let’s see what happens if we weaken compactness to other compact-like properties. We recall that a space is countably compact if each its open countable cover of has a finite subcover, or, equivalently, if each its infinite subset $$A$$ has an accumulation point $$x$$ (the latter means that each neighborhood of $$x$$ contains infinitely many points of the set $$B$$). A space is sequentially compact if each sequence in it contains a convergent subsequence. A space is feebly compact if each locally finite family of its nonempty open subsets is finite. It is well-known and easy to show that each (sequentially) compact space is countable compact and each countable compact space is feebly compact. A ($$T_1$$) completely regular space is feebly compact iff it is pseudocompact, that is when each continuous real-valued function on it is bounded.

If $$G$$ is a paratopological group which is a $$T_1$$ space and $$G\times G$$ is countably compact (in particular, if $$G$$ is sequentially compact) then $$G$$ is a topological group [RavRez]. On the other hand, we cannot weaken $$T_1$$ to $$T_0$$ here, because there exists a sequentially compact $$T_0$$ paratopological group which is not a topological group, see Example 2 from [Rav5]. Also we cannot weaken countable compactness of $$G\times G$$ to that of $$G$$ because under additional axiomatic assumptions there exists a functionally Hausdorff second countable paratopological group $$G$$ such that each power of $$G$$ is countably pracompact (and hence feebly compact), but $$G$$ is not a topological group, see [Rav4, Example 3]. On the other hand, by Proposition 3 from [Ra4] each feebly compact quasiregular paratopological group is a topological group. We recall that a space $$X$$ is quasiregular if each nonempty open subset $$A$$ of $$X$$ contains the closure of some nonempty open subset $$B$$ of $$X$$. In particular, each completely regular ($$T_1$$) paratopological group is a topological group.

On the other hand, there exists a pseudocompact seimitopogical group $$G$$ of period $$2$$, which is not a paratopological group, see [Kor1], [Kor2], see also [ArhTka, p.124-127]. On the other hand, Reznichenko in Theorem 2.5 of [Rez] showed that each semitopological group $$G\in \mathcal N$$ is a topological group, where $$\mathcal N$$ is a family of all pseudocompact spaces $$X$$ such that $$(X,X)$$ is a Grothendieck pair, that is if each continuous image of $$X$$ in $$C_p(Y)$$ has the compact closure in $$C_p(Y)$$. In particular, a pseudocompact space $$X$$ belongs to $$\mathcal N$$ provided $$X$$ has one of the following properties: countable compactness, countable tightness, separability, $$X$$ is a $$k$$-space, see [Rez].

References

[AlaSan] Ofelia T. Alas, Manuel Sanchis, Countably Compact Paratopological Groups, Semigroup Forum 74 (2007), 423-438.

[ArhTka] Alexander Arhangel'skii, Mikhail Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.

[BanRav2] Taras Banakh, Alex Ravsky, Each regular paratopological group is completely regular, Proc. Amer. Math. Soc. 145:3 (2017), 1373-1382.

[Bou] Ahmed Bouziad, Every Čech-analytic Baire semitopological group is a topological group, Proc. Amer. Math. Soc. 124 (1996), 953-959.

[Bra] N. Brand, Another note on the continuity of the inverse, Arch. Math 39, 241-245.

[Ell1] Robert Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), 119-125.

[Ell2] Robert Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc. 8 (1957), 372-373.

[Kor1] A. V. Korovin, Continuous actions of Abelian groups and topological properties in $$C_p$$-theory, Ph.D. Thesis, Moskow State University, Moskow (1990) (in Russian).

[Kor2] A. V. Korovin, Continuous actions of pseudocompact groups and the topological group axioms, Deposited in VINITI, #3734-D, Moskow (1990) (in Russian).

[Mon] D. Montgomery, Continuity in topological groups, Bull. Amer. Math. Soc. 42 (1936), 879-882.

[Pfi] Helmut Pfister, Continuity of the inverse, Proc. Amer. Math. Soc. 95 (1985), 312-314.

[Rav] Alex Ravsky, Paratopological groups I, Matematychni Studii 16:1 (2001), 37-48.

[Rav2] Alex Ravsky, *Paratopological groups II, Matematychni Studii 17:1 (2002), 93-101.

[Rav3] Alex Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis. - Lviv University, 2002 (in Ukrainian).

[Rav4] Alex Ravsky, Pseudocompact paratopological groups that are topological.

[Rav5] Alex Ravsky, Cone topologies of paratopological groups.

[RavRez] Alex Ravsky, Eugenii Reznichenko, The continuity of inverse in groups, International Conference on Functional Analysis and its Applications Dedicated to the 110th anniversary of Stefan Banach, Book of Abstracts, May 28-31, 2002, Lviv, 170-172.

[Rez] Eugenii Reznichenko, Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl. 59 (1994), 233-244.

[Rez2] Eugenii Reznichenko, Čech complete semitopological group are topological groups, (preprint).

[Tka2] Mikhail Tkachenko, Semitopological and paratopological groups vs topological groups, In: Recent Progress in General Topology III (K.P. Hart, J. van Mill, P. Simon, eds.), 2013, 803-859.

[Tka3] Mikhail Tkachenko, Axioms of separation in semitopological groups and related functors, Topology Appl., 161 (2013), 364-376.

[Wal] A. D. Wallace, The structure of topological semigroups, Amer. Math. Soc. Bull. 61 (1955), 95-112.

Hint: This is a starter which might enable research in the net. In the classic Counterexamples in Topology by L.A. Steen and J.A. Seebach, Jr. there is a general reference chart in the appendix listing $$61$$ properties of a total of $$143$$ topological spaces.

The following topological spaces have the wanted properties:

T2 spaces which are not completely regular:

• (47): An altered long line

• (60): Relatively prime integer topology

• (63): Countable complement extension topology

• (64): Smirnov's deleted sequence topology

• (66): Indiscrete rational extension of $$\mathbb{R}$$

• (67): Indiscrete irrational extension of $$\mathbb{R}$$

• (68): Pointed rational extension of $$\mathbb{R}$$

• (69): Pointed irrational extension of $$\mathbb{R}$$

• (72): Rational extension in the plane

• (74): Double origin topology

• (75): Irrational slope topology

• (76): Deleted diameter topology

• (78): Half-disk topology

• (79): Irregular lattice topology

• (80): Arens square

• (81): Simplified Arens square

• (88): Alexandroff plank

• (90): Tychonoff corkscrew

• (91): Deleted Tychonoff corkscrew

• (92): Hewitt's condensed corkscrew

• (94): Thomas's plank

• (96): Thomas's corkscrew

• (100): Minimal Hausdorff topology

• (113): Strong ultrafilter topology

• (125): Gustin's sequence space

• (126): Roy's lattice space

• (127): Roy's lattice subspace

• It is probably useful to know that searching among examples from this book is possible also in the online in the database called pi-base: topology.jdabbs.com/spaces?q=t_2%20%2B%20~Completely%20Regular – Martin Sleziak Feb 16 at 11:58
• @MartinSleziak: Interesting, thanks. I wasn't aware of it and checked the general reference chart in the old-fashioned style. :-) – Markus Scheuer Feb 16 at 13:36
• @MartinSleziak: Many thanks for granting the bounty. :-) – Markus Scheuer Feb 19 at 12:11
• I do not see how this list answers the question. As an experiment, I googled "An altered long line" and came up with nothing of value (mostly references to "Counterexamples in Topology"). – Moishe Kohan Feb 21 at 0:42
• @MoisheCohen: This list gives you the right candidates at hand. I said research in the net, which indicates considerably more effort than simple googling. The net is just one source of information, you might have other sources which you can consult. The point is, that reasearch can be done more efficiently when candidates of topological spaces are known. – Markus Scheuer Feb 21 at 8:00