# LCH topologies on Groups that are not group topologies

Ellis's 1957 paper on Locally Compact Transformation groups proves the following:

A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately) continuous, is a group topology.

In other words, LCH semitopological groups are topological groups.

I am interested in examples demonstrating obstructions to extending this result. Specifically, I want to know if there are nice' examples of LCH topologies on groups such that:

i) left multiplication is continuous but right is not

ii) left multiplication is continuous but the inverse map is not.

In the literature, I've found there is extensive work weakening the topological conditions. However, I've not seen examples of the above situations keeping LCH.

Any help finding such examples is greatly appreciated. Thanks!

• Here is an example of i) which is even homeomorphic to $\mathbb{R}^2$: mathoverflow.net/q/158416. But I don´t know if it classifies as ´nice´. Apr 8, 2016 at 18:50
• Of course, any example satisfying (i) will also satisfy (ii); if left multiplication and inverse are continuous then so is right multiplication, since $yx = (x^{-1}y^{-1})^{-1}$. Apr 8, 2016 at 20:09
• Yes. Indeed any two of (left multiplication, right multiplication, and inverse) being continuous implies the third. In practice one or the other might be easier to see...and for constructions of such monsters I wanted to ask about both. Good point nonetheless Nate. Apr 8, 2016 at 20:20
• Ramiro, thank you for the example. Nice' would be an overstatement for that little beasty though: nonmeasurable (additive hom) and addition twisted together... Still, this is the first example I've seen! Apr 9, 2016 at 17:00
• The "two arrows" space is compact, Hausdorff, first countable, non-metrizable and carries the structure of a left-topological group (isomorphic to the group of isometries of the unit circle). In place of "two arrows", just consider "two circles" space (which is homeomorphic to the "two arrows" space). Observe that the "two arrows" space cannot be homeomorphic to a topological group (being first countable and non-metrizable). Apr 13, 2016 at 13:13