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One service mathematics has rendered the 'Et moi, "0' si j'avait su
oomment en revenir. human race. It has put common sense back je n'y
serais point aile: ' Jules Verne where it belongs. on the topmost
shelf next to the dusty canister labelled 'discarded n- sense'. The
series is divergent; therefore we may be able to do something with
it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A
highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics
serve as tools for other parts and for other sciences. Applying a
simple rewriting rule to the quote on the right above one finds
such statements as: 'One service topology has rendered mathematical
physics .. .'; 'One service logic has rendered com puter science ..
.'; 'One service category theory has rendered mathematics .. .'.
All arguably true. And all statements obtainable this way form part
of the raison d'el: re of this series."
General topology is the domain ofmathematics devoted to the
investigation of the concepts of continuity and passage to a limit
at their natural level of generality. The most basic concepts of
general topology, that of a topological space and a continuous map,
were introduced by Hausdorffin 1914.
Oneofthecentralproblemsoftopologyisthedeterminationandinvestigation
of topological invariants; that is, properties ofspaces which are
preserved under homeomorphisms. Topological invariants need not be
numbers. Connectedness, compactness, andmetrizability, forexample,
arenon-numericaltopologicalinvariants.Dimen sional invariants, on
the otherhand, areexamplesofnumericalinvariants which take
integervalues on specific topological spaces. Part II ofthis book
is devoted to them. Topological invariants which take values in the
cardinal numbers play an especially important role, providing the
raw material for many useful coin" putations. Weight, density,
character, and Suslin number are invariants ofthis type. Certain
classes of topological spaces are defined in terms of topological
in variants. Particularly important examples include the metrizable
spaces, spaces with a countable base, compact spaces, Tikhonov
spaces, Polish spaces, Cech complete spaces and the symmetrizable
spaces."
One service mathematics has rendered the 'Et moi, "0' si j'avait su
oomment en revenir. human race. It has put common sense back je n'y
serais point aile: ' Jules Verne where it belongs. on the topmost
shelf next to the dusty canister labelled 'discarded n- sense'. The
series is divergent; therefore we may be able to do something with
it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A
highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics
serve as tools for other parts and for other sciences. Applying a
simple rewriting rule to the quote on the right above one finds
such statements as: 'One service topology has rendered mathematical
physics .. .'; 'One service logic has rendered com puter science ..
.'; 'One service category theory has rendered mathematics .. .'.
All arguably true. And all statements obtainable this way form part
of the raison d'el: re of this series."
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