Abstract
A proLie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected proLie groups is open. In fact this remains true for almost connected proLie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for proLie groups in the almost connected context.
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2000 Mathematics Subject Classification:
primary 22A05, 22E65; secondary 46A30

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MathSciNet:
MR2354??? 
Z'blattMATH:
pre05231332 
^{†}indicates author for correspondence 
References

N. Bourbaki, Groupes et algèbres de Lie, chapters 2–3 (Hermann, Paris, 1972).
MR573068

N. Bourbaki, Topologie générale, chapters 5–10 (Hermann, Paris, 1974).
MR1726872

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I Structure of topological groups. Integration theory, group representations (Academic Press, Inc., Publishers, New York, 1963).
MR156915

K. H. Hofmann, ‘On a category of topological groups suitable for a structure theory of locally compact groups’, Topology Proceedings 26 (2001–2002), 651–665.
MR2032841

K. H. Hofmann and S. A. Morris, The Structure of Compact Groups (De Gruyter Berlin, 1998).
MR1646190

K. H. Hofmann and S. A. Morris, ‘Projective limits of finite dimensional Lie groups’, Proc. London Math. Soc. 87 (2003), 647–676.
MR2005878

K. H. Hofmann and S. A. Morris, ‘The structure of abelian proLie groups’, Math. Z. 248 (2004), 867–891.
MR2103546

K. H. Hofmann and S. A. Morris, ‘Sophus Lie's third fundamental theorem and the adjoint functor theorem’, J. Group Theory 8 (2005), 115–133.
MR2115603

K. H. Hofmann and S. A. Morris, The Lie Theory of Connected ProLie Groups the Structure of ProLie Algebra, ProLie Groups and Locally Compact Groups (EMS Publishing House, Zürich, 2007).
MR2337107

K. H. Hofmann, S. A. Morris and D. Poguntke, ‘The exponential function of locally connected compact abelian groups’, Forum Math. 16 (2003), 1–16.
MR2034540

F. Burton Jones, ‘Connected and disconnected plane sets and the functional equation f(x+y)=f(x)+f(y)’, Bull. Amer. Math. Soc. 48 (1942), 115–120.
MR5906

Sh. Koshi and M. Takesaki, ‘An open mapping theorem on homogeneous spaces’, J. Aust. Math. Soc., Ser. A. 53 (1992), 51–54.
MR1164775

D. Montgomery and L. Zippin, Topological Transformation Groups (Interscience Publishers, New York, 1955).
MR73104

W. Roelcke and S. Dierolf, Uniform Structures on Topological Groups and their Quotients (McGrawHill, New York, 1981).
MR644485

H. Yamabe, ‘Generalization of a theorem of Gleason’, Ann. of Math. 58 (1953), 351–365.
MR58607

H. Yamabe, ‘On the conjecture of Iwasawa and Gleason’, Ann. of Math. 58 (1953), 48–54.
MR54613
