# Introduction to Mathematical Physics/Topological spaces

## Definition

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

${\displaystyle \lim u_{n}=u.}$

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

Definition:

A sequence ${\displaystyle u_{n}}$ of points of a topological space has a limit ${\displaystyle l}$ if each neighbourhood of ${\displaystyle l}$ contains the terms of the sequence after a certian rank.

## Distances and metrics

Definition:

A distance on an ensemble ${\displaystyle E}$ is an map ${\displaystyle d}$ from ${\displaystyle E\times E}$ into ${\displaystyle R^{+}}$ that verfies for all ${\displaystyle x,y,z}$ in ${\displaystyle E}$:

${\displaystyle d(x,y)=0}$ if and only if ${\displaystyle x=y}$. ${\displaystyle d(x,y)=d(y,x)}$. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$.

Definition:

A metrical space is a couple ${\displaystyle (A,d)}$ of an ensemble ${\displaystyle A}$ and a distance ${\displaystyle d}$ on ${\displaystyle A}$.

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 ${\displaystyle \pi }$ 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 ${\displaystyle (A,d)}$ a metrical space. A sequence ${\displaystyle x_{n}}$ of elements of ${\displaystyle A}$ is said a Cauchy sequence if ${\displaystyle \forall \epsilon >0\exists n,\forall (p,q),p>n,q>n,d(x_{p},x_{q})<\epsilon }$.

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 ${\displaystyle (A,d)}$ is said complete if any Cauchy sequence of elements of ${\displaystyle A}$ converges in ${\displaystyle A}$.

The space ${\displaystyle R}$ is complete. The space ${\displaystyle Q}$ of the rational number is not complete. Indeed the sequence ${\displaystyle u_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}}$ is a Cauchy sequence but doesn't converge in ${\displaystyle Q}$. It converges in ${\displaystyle R}$ to ${\displaystyle e}$, that shows that ${\displaystyle e}$ 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 ${\displaystyle E}$ 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 ${\displaystyle L^{2}}$ is a Hilbert space.

## Tensors and metrics

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

${\displaystyle ds^{2}=g_{ij}dx^{i}dx^{j}=g^{ij}dx_{i}dx_{j}}$

Covariant components ${\displaystyle x_{i}}$ can be expressed with respect to contravariant components:

${\displaystyle x_{i}=g_{ij}x^{j}}$

The invariant ${\displaystyle x^{i}y_{j}}$ can be written

${\displaystyle x^{i}y_{j}=x^{i}g_{ij}y^{j}}$

and tensor like ${\displaystyle a_{i}^{j}}$ can be written:

${\displaystyle a_{i}^{j}=g^{jk}a_{ik}}$

## Limits in the distribution's sense

Definition:

Let ${\displaystyle T_{\alpha }}$ be a family of distributions depending on a real parameter ${\displaystyle \alpha }$. Distribution ${\displaystyle T_{\alpha }}$ tends to distribution ${\displaystyle T}$ when ${\displaystyle \alpha }$ tends to ${\displaystyle \lambda }$ if:

${\displaystyle \forall \phi \in {\mathcal {D}},\lim _{\alpha \rightarrow \lambda }=}$

In particular, it can be shown that distributions associated to functions ${\displaystyle f_{\alpha }}$ verifying:

${\displaystyle f_{\alpha }(x)\geq 0}$

${\displaystyle \int f_{\alpha }(x)dx=1}$

${\displaystyle \forall a\geq 0,\lim _{\alpha \rightarrow \infty }\int _{|x|>a}f_{\alpha }(x)dx=0}$

converge to the Dirac distribution.

figdirac

Family of functions ${\displaystyle f_{\epsilon }}$ where ${\displaystyle f_{\epsilon }}$ is${\displaystyle 1/\epsilon }$ over the interval ${\displaystyle [0,\epsilon ]}$ et zero anywhere else convergesto the Dirac distribution when ${\displaystyle \epsilon }$ tends to zero.

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