1. Basic notions of topology

This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).

A topology on a set $S$ is a collection $\mathscr{T}$ of subsets containing both the empty set ∅ and the set $S$ such that $\mathscr{T}$ is closed under arbitrary unions and finite intersections, i.e.,

(i) if $U_a ∈ \mathscr{T}$ for all $a$ in an index set $A$, then $\bigcup_{a∈A}U_a ∈ \mathscr{T}$,
(ii) if $U_1, . . . ,U_n ∈ \mathscr{T}$, then $\bigcap_{i=1}^n U_i ∈ \mathscr{T}.$

The elements of $\mathscr{T}$ are called open sets. The set $S$ with a topology will be called a topological space.

If $A$ is a subset of a topological space $S$, then the subspace topology on $A$ is defined as $\mathscr{T}_A = \{U ∩ A |\ \ U ∈ \mathscr{T}\}.$

A neighborhood of a point $p$ in $S$ is an open set $U$ containing $p$.

A subcollection $\mathcal{B}$ of a topology $\mathscr{T}$ on a topological space $S$ is a basis for the topology $\mathscr{T}$ if given an open set $U$ and point $p\in U$, there is an open set $B ∈ \mathcal{B}$ such that $p ∈ B ⊂U$. We also say that $\mathcal{B}$ generates the topology $\mathscr{T}$ or that $\mathcal{B}$ is a basis for the topological space $S$. A collection $\mathcal{B}$ of open sets of $S$ is a basis if and only if every open set in $S$ is a union of sets in $\mathcal{B}$.

If $X,Y$ are topological spaces a function $f : X →Y$ is continuous if and only if the inverse image of any open set is open.
A continuous bijection $f : X →Y$ whose inverse is also continuous is called a homeomorphism.

Topologigal manifold

A topological manifold $M$ of dimension $n$ is a topological space with the following properties:
(i) $M$ is Hausdorff, that is, for each pair $p_1,p_2$ of distinct points of $M$ there exist neighborhoods $V_1,V_2$ of $p_1$ and $p_2$, respectively such that $V_1∩V_2=∅$.
(ii) Each point $p∈M$ possesses a neighborhood $V$ homeomorphic to an open subset $U$ of $R^n$.
(iii) $M$ satisfies the second countability axiom, that is, $M$ has a countable basis for its topology.

For an $n$-dimensional topological manifold each pair $(U, φ)$, where $U$ is an open subset of $M$ and $φ : U → φ(U ) ⊂ R^n$ is a homeomorphism of $U$ to an open subset of $R^n$ is called a coordinate map, chart or coordinate system and $U$ is a coordinate neighborhood.

For $p\in U,\ $ $φ( p)$ belongs to $R^n$, and therefore consists of $n$ real numbers that depend on $p$. Thus $φ( p)$ is of the form

$$ φ( p) = (x^1 ( p), x^2 ( p), . . . , x^n ( p)).$$

Smooth manifold

A map $\phi=(\phi^1,\ldots,\phi^m)\ $ from an open subset $U\subset R^n$ to $R^m$ is smooth on $U$ or belongs to $C^\infty(U)$ if all partial derivatives $$\frac{\partial^{\alpha_1+\ldots+\alpha_n} \phi^k} {\partial (x^1)^{\alpha_1}\ldots\partial (x^n)^{\alpha_n}}, \quad \text{where }\ \alpha_i \ \text{denote non-negative integers} $$ are continuous on $U$.

When two coordinate neighborhoods overlap, we have formulas for the associated coordinate change. The idea to obtain smooth manifolds is to choose a subcollection of coordinate neighborhoods so that the coordinate changes are smooth maps.

An $n$-dimensional $C^\infty$ or smooth manifold is a topological manifold of dimension $n$ and a family of coordinate charts $φ_α : U_α → R^n$ defined on open sets $U_α ⊂ M$, such that:

(i) the coordinate neighborhoods $U_\alpha$ cover $M$,

(ii) for each pair of indices $α, β$ such that $W := U_α \cap U_β \not= ∅,$ the overlap maps (transitions)

$φ_β ◦ φ_α^{-1} : φ_α (W ) → φ_β (W ),$
$φ_α ◦ φ_β^{-1} : φ_β (W ) → φ_α (W ),$

are $C^\infty$,

(iii) the family $A = \{(U_α , φ_α )\}$ is maximal with respect to (i) and (ii), meaning that if $φ_0 : U_0 → R^n$ is a chart such that $φ_0 ◦ φ^{-1}$ and $φ\circ φ_0^{-1}$ are $C^\infty$ for all $φ \in A$, then $(U_0 , φ_0 )$ is in $A$.

Any family $A = \{(U_α , φ_α )\}$ that satisfies (i) and (ii) is called a $C^∞$-atlas for $M$. If $A$ also satisfies (iii) it is called a maximal atlas or a differentiable or smooth structure.

Given any atlas $A = \{(U_α , φ_α )\}$ on $M$, there is a unique maximal atlas $\bar{A}$ containing it. In fact, we can take the set $\bar{A}$ of all maps that satisfy (ii) with every coordinate neighborhoods on $A$. Clearly $A ⊂ \bar{A}$, and one can easily check that $\bar{A}$ satisfies (i) and (ii). Also, by construction, $\bar{A}$ is maximal with respect to (i) and (ii). Two atlases are said to be equivalent if they define the same differentiable structure.

Smooth functions and maps.

By $C^\infty(M)$ we shall denote the family of smooth functions on a smooth manifold $M$, i.e., functions $f$, such that $f\circ\phi^{-1}$ is smooth on $\phi(U)\subset R^n$ for every coordinate chart $(U,\phi)$.

If $M$, $N$ are smooth manifolds, then a map $\psi: M\to N$ is smooth if for every pair of charts $(U, φ)$ on $M$ and $(V, χ )$ on $N$, the map $χ ◦ ψ ◦ φ^{−1}$ is smooth on $φ(\psi^{-1}(V)\cap U)\subset R^n.$

What's next?

Take a look at the notebook Examples of charts. Cartesian and spherical coordinates.