A bounded infinite subset S of the real numbers possesses at least
one point of accumulation p, i.e. p satisfies the property that given any
e > 0 there is an infinite sequence {p
}
of points of S with
|p - p
| < e.
Listing had examined connectivity in three dimensional Euclidean space but Betti extended his ideas to n dimensions. This is not as straightforward as it might appear since even in three dimensions it is possible to have a surface that cannot be reduced to a point yet closed curves on the surface can be reduced to a point. Betti's definition of connectivity left something to be desired and criticisms were made by Heegaard.
The idea of connectivity was eventually put on a completely rigorous basis by Poincaré in a series of papers Analysis situs in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself. Euler's convex polyhedra formula had been generalised to not necessarily convex polyhedra by Jonquières in 1890 and now Poincaré put it into a completely general setting of a p-dimensional variety V.
Also while dealing with connectivity Poincaré introduced the fundamental group of a variety and the concept of homotopy was introduced in the same 1895 papers.
A second way in which topology developed was through the generalisation of the ideas of convergence. This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
Cantor in 1872 introduced the concept of the first derived set, or set of limit points, of a set. He also defined closed subsets of the real line as subsets containing their first derived set. Cantor also introduced the idea of an open set another fundamental concept in point set topology.
Weierstrass in 1877 in a course of unpublished lectures gave a rigorous proof of the Bolzano- Weierstrass theorem which states
A bounded infinite subset S of the real numbers possesses at least one point of accumulation p, i.e. p satisfies the property that given any e > 0 there is an infinite sequence {p
|p - p
Hence the concept of neighbourhood of a point was introduced.
Hilbert used the concept of a neighbourhood in 1902 when he answered in the affirmative one of his own questions, namely
Is a continuous transformation group differentiable?
In 1906 Fréchet called a space compact if any infinite bounded subset contains a point of accumulation. However Fréchet was able to extend the concept of convergence from Euclidean space by defining metric spaces. He also showed that Cantor's ideas of open and closed subsets extended naturally to metric spaces.
Riesz, in a paper to the International Congress of Mathematics in Rome (1909), disposed of the metric completely and proposed a new axiomatic approach to topology. The definition was based on an set definition of limit points, with no concept of distance. A few years later in 1914 Hausdorff defined neighbourhoods by four axioms so again there were no metric considerations. This work of Riesz and Hausdorff really allows the definition of abstract topological spaces.
There is a third way in which topological concepts entered mathematics, namely via functional analysis. This was a topic which arose from mathematical physics and astronomy, brought about because the methods of classical analysis were somewhat inadequate in tackling certain types of problems. Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated.
Hadamard introduced the word 'functional' in 1903 when he studied linear functionals F of the form
F(f) = lim
f(x)g
where the limit is taken as n --> 
and the integral is from a to b. Fréchet
continued the development of functional by defining the derivative of a
functional in 1904.
Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series. Distance was defined via an inner product. Schmidt's work on sequence spaces has analogues in the theory of square summable functions, this work being done also in 1907 by Schmidt himself and independently by Fréchet.
A further step in abstraction was taken by Banach in 1932 when he moved from inner product spaces to normed spaces. Banach took Fréchet's linear functionals and showed that they had a natural setting in normed spaces.
Poincaré developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems. His study of autonomous systems
dx/dt = f(x,y), dy/dt = g(x,y)
involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier. The collection of methods developed by Poincaré was built into a complete topological theory by Brouwer in 1912.
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