Page 3 - Elementary Concepts of Topology

PREFACE

FEW BRANCHES of geometry have developed so rapidly and successfully
in recent times as topology, and rarely has an initially unpromising branch
of a theory turned out to be of such fundamental importance for such a
great range of completely different fields as topology. Indeed, today in
nearly all branches of analysis and in its far-reaching applications, topo-
logical methods are used and topological questions asked.
Such a wide range of applications naturally requires that the conceptual
structure be of such precision that the common core of the superficially
different questions may be recognized. It is not surprising that such an
analysis of fundamental geometrical concepts must rob them to a large
extent of their immediate intuitiveness—so much the more, when in the
application to other fields, as in the geometry of our surrounding space,
an extension to arbitrary dimensions becomes necessary.
While I have attempted in my Anschauliche Geometric to consider spatial
perception, here it will be shown how many of these concepts may be
extended and sharpened and thus, how the foundation may be given for a
new, self-contained theory of a much extended concept of space. Never-
theless, the fact that again and again vital intuition has been the driving
force, even in the case of all of these theories, forms a glowing example of
the harmony between intuition and thought.
Thus the following book is to be greeted as a welcome complement to
my Anschauliche Geometric on the side of topological systematization;
may it win new friends for the science of geometry.
DAVID HILBERT