# Introduction to Mathematical Physics/Topological spaces
## Definition

For us a topological space is a space where one has given a sense to:
$\lim u_{n}=u.$
Indeed, the most general notion of limit is expressed in topological spaces:
Definition:
A sequence $u_{n}$ of points of a topological space has a limit $l$ if each neighbourhood of $l$ contains the terms of the sequence after a certian rank.

## Distances and metrics

Definition:
A distance on an ensemble $E$ is an map $d$ from $E\times E$ into $R^{+}$ that verfies for all $x,y,z$ in $E$:
$d(x,y)=0$ if and only if $x=y$. $d(x,y)=d(y,x)$. $d(x,z)\leq d(x,y)+d(y,z)$.
Definition:
A metrical space is a couple $(A,d)$ of an ensemble $A$ and a distance $d$ on $A$.
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.
Definition:
Let $(A,d)$ a metrical space. A sequence $x_{n}$ of elements of $A$ is said a Cauchy sequence if $\forall \epsilon >0\exists n,\forall (p,q),p>n,q>n,d(x_{p},x_{q})<\epsilon$.
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.
Definition:
A metrical space $(A,d)$ is said complete if any Cauchy sequence of elements of $A$ converges in $A$.
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.
Definition:
A normed vectorial space is a vectorial space equipped with a norm.
The norm induced a distance, so a normed vectorial space is a topological space (on can speak about limits of sequences).
Definition:
A separated prehilbertian space is a vectorial space $E$ that has a scalar product.
It is thus a metrical space by using the distance induced by the norm associated to the scalar product.
Definition:
A Hilbert space is a complete separated prehilbertian space.
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:
$ds^{2}=g_{ij}dx^{i}dx^{j}=g^{ij}dx_{i}dx_{j}$
Covariant components $x_{i}$ can be expressed with respect to contravariant components:
$x_{i}=g_{ij}x^{j}$
The invariant $x^{i}y_{j}$ can be written
$x^{i}y_{j}=x^{i}g_{ij}y^{j}$
and tensor like $a_{i}^{j}$ can be written:
$a_{i}^{j}=g^{jk}a_{ik}$

## Limits in the distribution's sense

Definition:
Let $T_{\alpha }$ be a family of distributions depending on a real parameter $\alpha$. Distribution $T_{\alpha }$ tends to distribution $T$ when $\alpha$ tends to $\lambda$ if:
$\forall \phi \in {\mathcal {D}},\lim _{\alpha \rightarrow \lambda }<T_{\alpha },\phi >=<T,\phi >$
In particular, it can be shown that distributions associated to functions $f_{\alpha }$ verifying:
$f_{\alpha }(x)\geq 0$
$\int f_{\alpha }(x)dx=1$
$\forall a\geq 0,\lim _{\alpha \rightarrow \infty }\int _{|x|>a}f_{\alpha }(x)dx=0$
converge to the Dirac distribution.
figdirac
Figure figdirac represents an example of such a family of functions.