After reading [this MO post](https://mathoverflow.net/questions/104575/under-exactly-what-extra-conditions-if-any-is-a-connected-hausdorff-manifold), I am wondering: 


**Is every (connected) Hausdorff Banach manifold a regular space?**


Though unjustified, page 53 of [this paper](http://www.maik.ru/full/rusmath/97/10/rusmath10_97p49full.pdf) nonchalantly states: "Note that a Hausdorff Banach manifold X is a regular space."


But does anyone know of a proof of this statement (or a counterexample)?


Of course, the real difficulty arises in proving the statement for the infinite-dimensional version, since such a Banach manifold will *not* be locally compact.




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**Follow-up:** Now that Theo Buehler has kindly pointed to a counterexample (i.e. a connected Hausdorff Banach manifold which is *not* regular) perhaps it will give someone an idea about how to tackle [the question](https://mathoverflow.net/questions/104575/under-exactly-what-extra-conditions-if-any-is-a-connected-hausdorff-manifold) that provided the inspiration for this one.



Non-regular Connected Hausdorff Banach Manifold
Apparently the answer is **no**, not every connected Hausdorff Banach manifold is regular, not even when it is modeled on a separable Hilbert space.


I quote (verbatim) from J. Margalef-Roig, E. Outerelo-Dominguez, *Differential Topology*, North Holland Mathematics Studies 173, 1992, page 44f.



> 
> It is well known the result of General Topology that every
>  Hausdorff locally compact topological space satisfies the
>  Tychonoff axiom [M-O-P, V.2, pg. 231]. By this and the Riesz's
>  theorem every Hausdorff locally finite dimensional differentiable
>  manifold satisfies the Tychonoff axiom. This last affirmation is not
>  true for arbitrary Hausdorff differentiable manifolds. In
>  [M.O.1] there is an example of a Hausdorff connected
>  differentiable manifold $X$ of class $\infty$, such that $\partial X = \emptyset$, $X$ is not
>  regular and $X$ admits an atlas whose charts are modelled over an
>  infinite dimensional separable real Hilbert space.
> 
> 
> 


They continue to add the regularity hypothesis in their results whenever it is needed.


The cited references are:


* [M.O.P.] MARGALEF, J.-OUTERELO, E.-PINILLA, J.L.: Topologia,
I-V. Alhambra, Madrid 1975, 79, 79, 80 and 1982.
* [M.O.1] MARGALEF, J.-OUTERELO, E.: Una variedad diferenciable
de dimension infinita, separada y no regular.
Rev. Mat. Hisp.-Am, IV, V.42, 1982, 51-55. [(QuickView link)](http://tinyurl.com/8kb3m45).




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**Edit:** As was pointed out by Benjamin Dickman in the comments, the example also appears in English in A. Kriegl, P.W. Michor, *[The convenient setting of global analysis](http://books.google.com/books?id=s7fPYRqhXEUC)*, AMS (1997), [**27.6** Non-regular manifold](http://books.google.com/books?id=s7fPYRqhXEUC&pg=PA266), page 266. The book is available as [a pdf file](http://www.mat.univie.ac.at/~kriegl/Skripten/apbook.pdf) on Kriegl's homepage.


