Introduction to Mathematical Physics/Topological spaces

Definition
For us a topological space is a space where one has given a sense to:

Indeed, the most general notion of limit is expressed in topological spaces:

Distances and metrics
To each metrical space can be associated a topological space. In this text, all the topological spaces considered are metrical space. In a metrical space, a converging sequence admits only one limit (the toplogy is separated ).

Cauchy sequences have been introduced in mathematics when is has been necessary to evaluate by successive approximations numbers like $$\pi$$ that aren't solution of any equation with inmteger coeficient and more generally, when one asked if a sequence of numbers that are ``getting closer'' do converge.

Any convergent sequence is a Cauchy sequence. The reverse is false in general. Indeed, there exist spaces for wich there exist Cauchy sequences that don't converge.

The space $$R$$ is complete. The space $$Q$$ of the rational number is not complete. Indeed the sequence $$u_n=\sum_{k=0}^n\frac{1}{k!}$$ is a Cauchy sequence but doesn't converge in $$Q$$. It converges in $$R$$ to $$e$$, that shows that $$e$$ is irrational.

The norm induced a distance, so a normed vectorial space is a topological space (on can speak about limits of sequences).

It is thus a metrical space by using the distance induced by the norm associated to the scalar product.

The space of summable squared functions $$L^2$$ is a Hilbert space.

Tensors and metrics
If the space $$E$$ has a metrics $$g_{ij}$$ then variance can be changed easily. A metrics allows to measure distances between points in the space. The elementary squared distance between two points $$x_i$$ and $$x_i+dx_i$$ is:

Covariant components $$x_i$$ can be expressed with respect to contravariant components:

The invariant $$x^iy_j$$ can be written

and tensor like $$a_i^j$$ can be written:

Limits in the distribution's sense
In particular, it can be shown that distributions associated to functions $$f_\alpha$$ verifying:

converge to the Dirac distribution.



Figure figdirac represents an example of such a family of functions.