spacetime.tex

\chapter{Spacetime}

\section{Introduction}

The fundamental notion of special relativity is that of an \emph{event}.
In pre-relativity, things located at a specific point in space and time,
for example the start of the Saturn V rocket launching the Apollo 11
spaceflight, are described by a unique time in Newton's absolute time and
a unique place in Euclidean absolute space. The notion of an event combines
these two qualities into one, which is essential for special relativity
as space and time by themselves lose their absoluteness.

\emph{Spacetime} is the set of all possible events. Any event that can
be imagined to happen is an element or a
\emph{point} in spacetime. Mathematically, spacetime has more
structure than simply being a set: A point in this set, that is an event, is
usually described by four scalars, one time and three space coordinates,
making spacetime four-dimensional. For a general set, however, there is no
well-defined notion of coordinates or dimension. The theory of manifolds, which
will be presented in the next section, is the correct mathematical setting in
which notions like coordinates and dimension make sense.

\section{Manifolds}
\label{sec:manifolds}

An \emph{$n$-dimensional chart $x = (x, U)$} on a set $M$, whose elements we call
\emph{points}, is an injective map
$x = (x_1, \dotsc, x_n)\colon U \to \set R^n$ onto an open subset of $\set R^n$ defined on a subset
$U$ of $M$, the \emph{domain of definition of $x$}. Given a point $p$ lying in the
domain of definition $U$, the $n$ scalars $x_1(p)$, \dots, $x_n(p)$ are the
\emph{coordinates of $p$ with respect to the chart $x$}. A
family $\Set{(x_i, U_i)}_{i \in I}$ of charts on $M$ \emph{covers} $M$ if $M \subseteq
\bigcup_{i \in I} U_i$ that is if every point $p$ of $M$ lies in at least domain
of definition of the charts $x_i$. In order to be able to describe events in
spacetime by four coordinates, we postulate that \emph{there is a distinguished
family of four-dimensional charts that cover spacetime}.

Given two $n$-dimensional charts $(x, U)$ and $(y, V)$ of a set $M$, the map
\[
  y \circ x^{-1}|x(U \cap V)\colon x(U \cap V) \to y(U \cap V)
\]
is called the
\emph{coordinate transformation from $x$ to $y$}. If this map is a diffeomorphism
between open subsets of $\set R^n$, the charts $x$ and $y$ are said to be
\emph{compatible}.
An \emph{$n$-dimensional atlas of $M$} is a family of pairwise compatible
$n$-dimensional charts of $M$ that covers $M$. In order to be able to employ 
analytic methods, we extend our postulate above by postulating that
\emph{there is a distinguished four-dimensional atlas of spacetime}.

Let $\mathfrak A$ be any $n$-dimensional atlas of a set $M$, and let $x$ and $y$
be two arbitrary $n$-dimensional charts of $M$. If $x$ and $y$ are each compatible
with each chart in $\mathfrak A$, they are compatible with each other. Therefore,
every atlas $\mathfrak A$ can be uniquely enlarged to a \emph{maximal atlas} in
which it is contained by adding all $n$-dimensional charts to $\mathfrak A$ that
are compatible with each in chart in $\mathfrak A$. We also say that the maximal
atlas is \emph{generated} by the charts contained in $\mathfrak A$.

An \emph{$n$-dimensional premanifold $M$} is a set $M$ together
with a maximal $n$-dimensional atlas $\mathfrak U^\infty(M)$ of $M$. The set of
all charts $(x, U)$ in $\mathfrak U^\infty(M)$ that contains a given point $p \in M$ is 
denoted by $\mathfrak U^\infty(p, M)$. A \emph{chart of $M$} is a chart in the
maximal atlas $\mathfrak U^\infty(M)$.
An \emph{atlas of the premanifold $M$} is any atlas of $M$ which is contained in
the maximal atlas $\mathfrak U^\infty(M)$. With these terms, we can say that
\emph{spacetime is a four-dimensional premanifold}.

Using the atlas of a premanifold $M$, one can define the notion of neighborhoods
of points on $M$: A subset $G$ of the underlying set of $M$ is an \emph{open subset
of $M$} if $x(U \cap G)$ is an open subset for each chart $(x, U)$ of $M$. The
system of these open subsets of $M$ is a topology (see~\prettyref{sec:topological_spaces})
on the underlying set of $M$, the \emph{canonical topology of $M$}, so that $M$
is a topological space in a canonical way.

For any chart $(x, U)$ of the $n$-dimensional premanifold, a subset $V$ of $U$
is open in $M$ if and only if $x(V)$ is open in $\set R^n$. In particular, $U$ is open.
The map $x\colon U \to \set R^n$ is continuous in the sense of maps between
topological spaces.

Generally, the so-defined topology is ill-behaved, however;
points may not be distinguishable by the topology and one may need infinitely
many charts to cover even small neighborhoods of points. Excluding these cases
by adding technical conditions on the underlying topological space leads to the
final definition:
\begin{dfn}
  An \emph{$n$-dimensional manifold $M$} is an $n$-dimensional premanifold $M$
  whose underlying topological space is a paracompact Hausdorff space
  (see~\prettyref{sec:paracompactness}).
\end{dfn}
We extend our postulate from above by postulating that \emph{spacetime is a
four-dimensional manifold}.

The most basic example of an $n$-dimensional manifold is the $n$-dimensional 
cartesian space $\set R^n$. Its canonical atlas is the maximal atlas which is generated
by the identity map $\id_{\set R^n}\colon \set R^n \to \set R^n$ viewed
as an $n$-dimensional chart. The topology defined by this atlas is, of course,
the canonical topology of $\set R^n$, which is the topology of a paracompact
Hausdorff space.

Requiring that the underlying topological space of spacetime is a paracompact
Hausdorff space has a number of pleasant consequences: For example, the underlying
topological spaces of manifolds are normal, which follows from \prettyref{prop:paracompact_spaces}.
Furthermore, we have:
\begin{prop}
  The underlying space of an $n$-dimensional manifold $M$ is locally compact.
\end{prop}
(For the definition of local compactness, see \prettyref{sec:locally_compact}.)

\begin{proof}
  For $p \in M$ choose a chart $(x, U) \in \mathfrak U^\infty(p, M)$. By openness
  of $x(U)$ in $\set R^n$, there exists an $\epsilon > 0$ such that $U_\epsilon(x(p))
  \subseteq x(U)$. We claim that
  \[
    K \coloneqq x^{-1}(\overline{U_{\frac \epsilon 2}(x(p))})
  \]
  is a compact neighborhood of $p$:
  
  The subset $K$ is a neighborhood of $p$ as
  $p \in x^{-1}(U_{\frac \epsilon 2}(x(p))) \subseteq K$ and
  because $x^{-1}(U_{\frac 1 2 \epsilon}(x(p)))$ is an open subset of $M$.

  To show that $K$ is compact, let $(U_i)_{i \in I}$ be a family of open subsets
  of $U$ that cover $K$. By definition of the topology of $M$, the images $(x(U_i))_{i \in I}$
  form an open cover of the compact space $\overline{U_{\frac \epsilon 2}(x(p))}$.
  Thus there is a finite subset $J \subset I$ such that $(x(U_i))_{i \in J}$ is
  an open cover of $\overline{U_{\frac \epsilon 2}(x(p)}$. It follows that
  $(U_i)_{i \in J}$ is a finite open cover of $K$.
\end{proof}

Often, we have to restrict our attention to small pieces (that is, open subsets)
of a given $n$-dimensional manifold $M$, for example spacetime. This can be
done as follows: Let $G$ be an open subspace of $M$. Let $\mathfrak U^\infty(G)$
be the unique maximal $n$-dimensional atlas of $G$ that contains the atlas
\[
  \Set{(x|U \cap G, U \cap G) : (x, U) \in \mathfrak U^\infty(M)}.
\]
Then $G$ becomes an $n$-dimensional manifold itself, whose underlying topological
space is the subspace $G$ of $M$. An $n$-dimensional manifold of this form is
called an \emph{open submanifold of $M$}. The domains of definitions of the charts of
$M$ are open submanifolds of $M$.

\section{Functions on a manifold}
\label{sec:functions}

As we have postulated, we can ``measure'' each event in spacetime by giving the
scalar values of four coordinates (after choosing a chart). Each coordinate can
be thought of as a scalar field that assigns to each point (in the domain of
definition of its chart) a scalar value. This notion is generalised as follows:

\begin{dfn}
  A \emph{(smooth) function $\phi\colon M \to \set R$} on an $n$-dimensional
  manifold $M$ is a mapping $\phi$ from the underlying set of $M$ to the reals
  such that for each chart $(x, U)$ of $M$ the map
  \[
    \phi \circ x^{-1}\colon x(U) \to \set R
  \]
  is smooth.
\end{dfn}

In order to show that a map $\phi\colon M \to \set R$ is a function in the above
sense, it suffices that
for each point $p \in M$ there exists a chart $(x, U) \in \mathfrak U^\infty(p, M)$
such that $\phi \circ x^{-1}\colon x(U) \to \set R$ is smooth.

Every function is continuous with respect to the underlying
topologies of $M$ and $\set R$, respectively. Let $\phi$ be a function on $M$
and $G$ an open submanifold. The \emph{restriction $\phi|G$ of $\phi$ to $G$},
given by
\[
  \phi|G\colon G \to M, p \mapsto \phi(p)
\]
is a function on the manifold $G$. For any chart $(x, U)$ of $M$, the coordinate
functions $x_1$, \dots, $x_n\colon U \to M$ are smooth functions on the open
submanifold $U$.

All functions on $M$ form an algebra (over the reals), denoted by
$\mathcal C^\infty(M)$, where addition and multiplication are defined point-wise.
The function with constant value $c \in \set R$ is often denoted by $\underline c$.
The sets of functions of the open submanifolds of $M$ fulfill the \emph{sheaf
condition}, that is for every open cover $(U_i)_{i \in I}$ of $M$ and functions
$\phi_i \in \mathcal C^\infty(U_i), i \in I$ one has
\[
  \left(\forall i, j \in I : \phi_i|U_i \cap U_j = \phi_j|U_i \cap U_j\right)
  \implies \exists! \phi \in \mathcal C^\infty(M) \, \forall i \in I :
  \phi|U_i = \phi_i.
\]
In other words, we can uniquely glue functions along open submanifolds.

If $\Phi\colon \set R \to \set R$ is any smooth function between the reals, 
the composition $\Phi \circ \phi\colon M \to \set R$ is a function on $M$
whenever $\phi\colon M \to \set R$ is a function.

A manifold possesses many functions in a sense made precise by the following theorem,
which relies essentially on the paracompactness of the manifold (for the notion
of the support $\supp$ of a function, see \prettyref{sec:continuity}):
\begin{thm}
  \label{thm:paracompact_manifold}
  Every open cover $(U_i)_{i \in I}$ of a manifold $M$ has a
  \emph{subordinate
  partition $(\lambda_i)_{i \in I}$ of unity}, which is a family of functions
  $\lambda_i\colon M \to \set R$ with the following properties:

  \paragraph{Range}
  For all $i \in I$ and $p \in M$, one has $0 \leq \lambda_i(p)
  \leq 1$.
  
  \paragraph{Support}
  For all $i \in I$, one has $\supp \lambda_i \subseteq U_i$.
  
  \paragraph{Local finiteness}
  Every $p \in M$ possesses a neighborhood $G \in \mathfrak U^0(p, M)$ such that there
  are only finitely many $i \in I$ with $\supp \lambda_i \cap G \neq \emptyset$.
  
  \paragraph{Normalization}
  For every $p \in M$, the equality $\sum_{i \in I} \lambda_i(p) \equiv 1$ holds.
\end{thm}

The proof relies on the following lemma, which is also of independent interest:
\begin{lem}
  \label{lem:paracompact_manifold}
  Let $K$ be a compact subspace of an $n$-dimensional manifold $M$. For any open neighborhood
  $G$ of $K$ in $M$, there exists a function $\phi\colon M \to \set R$ with
  \begin{align}
    \label{eq:paracompact_manifold}
    \forall p \in M: \phi(p) & \ge 0, &
    \forall p \in K: \phi(p) & > 0, &
    \supp \phi & \subseteq G.
  \end{align}
\end{lem}

\begin{proof}[Proof of \prettyref{lem:paracompact_manifold}]
  For any $p \in K$ choose a chart $(x, U) \in \mathfrak U^\infty(p, M)$. 
  By the openness of $x(U)$ in $\set R^n$, there exists an $\epsilon > 0$ with
  $U_\epsilon(x(p)) \subseteq x(U \cap G)$. Choose a bump function $\psi\colon \set R^n \to
  \set R$ (see \prettyref{sec:bump_functions}) such that $\psi(u) \ge 0$ for all
  $u \in \set R^n$, $\psi(u) = 1$ for
  $\norm u \leq \frac \epsilon 3$ and $\psi(u) = 0$ for $u \ge \frac {2 \epsilon} 3$.
  By the sheaf condition, 
  \[
    \phi_p\colon M \to \set R, q \mapsto \begin{cases}
      \psi(x(q) - x(p)) & \text{if $q \in U$}, \\
      0 & \text{if $q \in M \setminus x^{-1}(\overline{U_{\frac {2 \epsilon} 3}(x(p))})$}
    \end{cases}
  \]
  defines a function on $M$ as $U$ and $M \setminus x^{-1}(\overline{U_{\frac {2 \epsilon} 3}(x(p))})$
  form an open cover of $M$. This function has the properties
  \begin{align*}
    \forall q \in M: \phi_p(q) & \ge 0, &
    \phi_p(p) & > 0, &
    \supp \phi_p & \subseteq G.
  \end{align*}
  
  The open subsets $U_p \coloneqq \Set{q \in M : \phi_p(q) > 0}$ with $p \in K$ cover $K$. By
  compactness of $K$, there exists a finite subset $A \subseteq K$ such that
  $K$ is covered by $(U_p)_{p \in A}$. By construction, the function
  $\phi = \sum_{p \in A} \phi_p$
  fulfills \prettyref{eq:paracompact_manifold}.
\end{proof}

\begin{proof}[Proof of \prettyref{thm:paracompact_manifold}]
  For every $p \in M$ choose by local compactness of $M$ a relatively compact neighborhood $G_p
  \in \mathfrak U(p, M)$. By the covering property, there exists an $i \in I$ with
  $p \in U_i$. The intersection $G_p \cap U_i$ is again relatively compact in $M$,
  so we may assume that already $G_p \subseteq U_i$. So  
  $(G_p)_{p \in M}$ is a refinement of the cover $(U_i)_{i \in I}$.
  Let $(V_j)_{j \in J}$ be a locally finite refinement of $(G_p)_{p \in M}$. In
  particular, $(V_j)_{j \in J}$ is a locally finite refinement of $(U_i)_{i \in I}$
  and each $V_j$ is relatively compact in $M$. By the shrinking lemma, \prettyref{prop:shrinking_lemma},
  and the normality of the underlying topological space of $M$, there exists
  another open cover $(V'_j)_{j \in J}$ of $M$ with
  $\overline{V'_j} \subseteq V_j$ for all $j \in J$.
  
  For every $j \in J$, choose by compactness of $\overline{V'_j}$ and
  \prettyref{lem:paracompact_manifold} a function
  $\phi_j \in \mathcal C^\infty(M)$ with
  \begin{align*}
    \forall p \in M: \phi_j(p) & \ge 0, &
    \forall p \in V'_j: \phi_j(p) & > 0, &
    \supp \phi_j \subseteq V_j.
  \end{align*}
  By the local finiteness of the open cover $(V_j)_{j \in J}$, the sum
  $\phi \coloneqq \sum_{j \in J} \phi_j$ is locally a finite sum and thus defines
  a function on $M$ by the sheaf condition. By the covering property of
  $(V'_j)_{j \in J}$,
  one has $f(p) > 0$ for all $p \in M$.
  
  As $(V_j)_{j \in J}$ is a refinement of the cover $(U_i)_{i \in I}$, there exists a
  map $\alpha\colon J \to I$ with $V_j \subseteq U_{\alpha(i)}$ for all $j \in J$.
  For each $i \in I$ set $J_i \coloneqq \alpha^{-1}(i)$, so $(J_i)_{i \in I}$ becomes
  a partition of $I$. For all $i \in I$, set
  $U'_i \coloneqq \bigcup_{j \in J_i} V_j \subseteq U_i$ and finally
  \[
    \lambda_i = \sum_{j \in J_i} \frac{\phi_i}{\phi}.
  \]
  By a similar argument as above, the sum on the right hand side is locally finite
  and, thus, $\lambda_i$ is a function on $M$ with $\lambda_i(p) \ge 0$ for all
  $p \in M$.
  
  By construction, $\supp \lambda_i \subseteq \bigcup_{j \in J_i} V_j = U'_i \subseteq
  U_i$, which proves the support axiom of a partition of unity. The covering
  $(U'_i)_{i \in I}$ is locally finite as the covering $(V_j)_{j \in J}$ is locally
  finite; this uses the disjointness of the $J_i$. From $\supp \lambda_i \subseteq U'_i$
  for all $i \in I$, the local finiteness axiom follows. By construction,
  $\sum_{i \in I} \lambda_i \equiv 1$, which is the normalization axiom of a
  partition of unity.
  From this, the range axiom follows as we already know that $\lambda_i(p) \ge 0$
  for all $p \in M$.
\end{proof}

The existence of partitions of unity on manifolds implies that functions can
be extended in the following sense:

\begin{cor}
  Let $\phi$ be a function defined on an open
  neighborhood $G$ of a point $p$
  in a manifold $M$. Then there exists a function $\widehat \phi \in \mathcal C^\infty(M)$
  with $\supp \widehat\phi \subseteq G$ and
  such that $\widehat \phi$ coincides with $\phi$ on a neighborhood $U \subseteq G$
  of $p$ in $M$.
\end{cor}
The function $\widehat \phi$ is called an \emph{extension of $\phi$
by zero away from $p$}.

\begin{proof}
  By local compactness (see \prettyref{prop:locally_compact}),
  there exists a compact neighborhood $K \in \mathfrak U(p, G)$.
  Choose a partition $(\lambda, \mu)$ of unity subordinate to the open cover
  $(G, M \setminus K)$ of $M$. By the sheaf condition,
  \[
    \hat\phi\colon M \to \set R, p \mapsto \begin{cases}
      \lambda(p) \cdot \phi(p) & \text{if $p \in G$} \\
      0 & \text{if $p \in M \setminus \supp \lambda$}
    \end{cases}
  \]
  is a well-defined function on $M$ with $\supp \hat\phi \subseteq G$ and
  which coincides with $\phi$ on $K$.
\end{proof}

For any two points $p$, $q \in M$ with $p \neq q$, there exists an open
neighborhood $G \in \mathfrak U^0(p, M)$ with $q \notin G$. Extending the
constant function $\underline 1|G$ by zero away from $p$ yields a function,
which is $1$ on $p$ and $0$ on $q$. Thus, we have
\begin{equation}
  \label{eq:point_separation}
  \forall p, q \in M : p \neq q \implies
  \exists \lambda \in \mathcal C^\infty(M) : \lambda(p) = 1, \lambda(q) = 0.
\end{equation}

If we denote by $p^*$ for all $p \in M$ the algebra homomorphism
\begin{equation}
  p^*\colon \mathcal C^\infty(M) \to \set R, \phi \mapsto \phi(p),
\end{equation}
we can reformulate \prettyref{eq:point_separation} by saying that the map $p \mapsto p^*$ is injective,
that is the algebra of functions \emph{separate points}. The algebra of functions
on spacetime is therefore a full set of observables: for any two distinct events
there is a function that takes different values on both events.

\section{Morphisms}
\label{sec:morphisms}

A physical body traces out a curve of events in spacetime $M$, namely those events
where an observer meets the physical body, its \emph{world line}. If the physical
body carries a clock with it, each point of its world line is parametrized by
a scalar, the clock's time measured at that event. In other words, the path of
the physical body in spacetime together with its clock defines a map $J \to M$, 
where $J$ is an (open) interval. Both the domain and the target of this map is a 
manifold, where $J$ is viewed as an open submanifold of the reals. A physical
body does not jump through spacetime, so the map will be continuous.
In order to employ analytical methods, it is sensible to assume moreover that
this map is a morphism according to the following definition:

\begin{dfn}
  A \emph{morphism $f\colon M \to N$} between two manifolds $M$ and $N$ is a
  continuous map $f\colon M \to N$ such that for each pair of charts
  $(x, U) \in \mathfrak U^\infty(M)$ and $(y, V) \in \mathfrak U^\infty(N)$ the
  composition
  \[
    y \circ f \circ x^{-1}|x(f^{-1}(V) \cap U)\colon x(f^{-1}(V) \cap U) \to y(V) 
  \]
  is a smooth map (between open subsets of cartesian spaces).
\end{dfn}

In accordance with our above wording, a \emph{curve $\alpha\colon J \to M$} is a
morphism where $J$ is an open interval viewed as a submanifold of $\set R$. Thus
a world line of a physical body becomes a curve in this sense
by endowing it with a clock.

A map $f\colon M \to N$ between manifolds is a morphism if and only if
for all functions $\psi \in \mathcal C^\infty(N)$ the \emph{pullback of $\psi$
by $f$}, given by
\begin{equation}
  f^{-1}\psi\colon M \to \set R, p \mapsto \psi(f(p)),
\end{equation}
is a function on $M$. (The pullback itself is a map
\begin{equation}
  f^{-1}\colon \mathcal C^\infty(N) \to \mathcal C^\infty(M)
\end{equation}
of algebras.)

Thus, a map $\phi\colon M \to \set R$ defined on a manifold $M$ is a
morphism if and only if it is a function. Further, a map $f\colon G \to H$ between
open subsets of cartesian spaces is a morphism if and only if it is a smooth
map in the sense of calculus.

The identity $\id_M\colon M \to M$ of $M$ is a morphism. The composition $g \circ f$
of two morphisms $f\colon M \to N$ and $g\colon N \to P$ between manifolds is
again a morphism. The manifolds together with the morphisms between them thus
form a \emph{category}. A morphism $f\colon M \to N$ between manifolds that
is bijective and whose inverse $f^{-1}\colon N \to M$ is again a morphism is
called a \emph{diffeomorphism between $M$ and $N$}. In a categorical sense,
diffeomorphisms are exactly the isomorphisms.

The \emph{inclusion $i\colon U \to M, p \mapsto p$} of an open submanifold $U$
of $M$ is a morphism. Thus, the restriction $f|U = f \circ i$ of a morphism
$f\colon M \to N$ to $U$ is again a morphism.

\section{Product manifolds}

When investigating possible universe, that is possible spacetimes, we will have
to construct manifolds. One important construction, which constructs a manifold
out of simpler ones is the product. Let $M$ and $N$ be two manifolds of dimensions
$m$ and $n$, respectively. Given any chart $(x, U) \in \mathfrak U^\infty(M)$
and any chart $(y, V) \in \mathfrak U^\infty(N)$, the map
\[
  x \times y\colon U \times V \to \set R^{m + n}, (p, q) \mapsto (x(p), y(q))
\]
defines an $(m + n)$-dimensional chart on the cartesian product
\[
  M \times N \coloneqq \Set{(p, q) : p \in M, q \in N}.
\]

Let $\mathfrak U^\infty(M \times N)$ be the unique maximal atlas of $M \times N$
that contains the atlas
\[
  \Set{(x \times y, U \times V): (x, U) \in \mathfrak U^\infty(M),
    (y, V) \in \mathfrak U^\infty(N)}.
\]
Then $M \times N$ becomes an $(m + n)$-manifold, whose underlying topological 
space is the product topological space of $M$ and $N$.

\begin{dfn}
  The manifold $M \times N$ is called the \emph{product of the manifolds $M$
  and $N$}.
\end{dfn}

The two canonical projection maps $\pr_1\colon M \times N \to M, (p, q) \mapsto
p$ and $\pr_2\colon M \times N \to N, (p, q) \mapsto q$ are morphisms.

As simple example of a product manifold is given by $\set R^n$. More precisely,
for any $p$, $q$ with $p + q = n$, the map
\[
  \set R^p \times \set R^q \to \set R^n, ((u_1, \dotsc, u_p), (v_1, \dotsc, v_q))
  \mapsto (u_1, \dotsc, u_p, v_1, \dotsc, v_q)
\]
is a diffeomorphism. (Here, the left hand side is, of course, endowed with the
structure of a product manifold.)

One example of a product manifold in classical physics is given by Galilean
spacetime; it is the product $L \times Q$ of the one-dimensional manifold
$L$ of absolute time and the three-dimensional manifold $Q$ of Galilean space.

\section{Problems}

\begin{xca}
  Prove that every $n$-dimensional atlas of a set $M$ is contained in a unique
  maximal $n$-dimensional atlas of $M$.
\end{xca}

\begin{xca}
  Prove that the system of open subsets of a premanifold $M$ is a topology on
  the underlying set of $M$.
\end{xca}

\begin{xca}
  Let $(x, U)$ be a chart of an $n$-dimensional premanifold $M$. Show that a 
  subset $V$ of $U$ is open in $M$ if and only if $x(V)$ is open in $\set R^n$.
  Conclude that $x\colon U \to \set R^n$ is a continuous map.
\end{xca}

\begin{xca}
  Prove that an open submanifold of a manifold is in fact a manifold.
\end{xca}

\begin{xca}
  Prove the following:
  Let $M$ be a manifold and $\phi\colon M \to \set R$ a map. Assume that for each
  point $p \in M$ there exists a chart $(x, U) \in \mathfrak U^\infty(p, M)$
  such that $\phi \circ x^{-1}\colon x(U) \to \set R$ is smooth. Then $\phi$
  is a (smooth) function.
\end{xca}

\begin{xca}
  Prove that every function $f$ on an $n$-dimensional manifold $M$ is a
  continuous map $f\colon M \to \set R$ for the underlying topologies of $M$ and
  $\set R$.
\end{xca}

\begin{xca}
  Prove that the functions on a manifold $M$ form a \emph{sheaf of algebras},
  that is prove that $\mathcal C^\infty(U)$ is closed under constants,
  addition and multiplication for every open subset $U$ of $M$, that the restrictions
  $\mathcal C^\infty(U) \to \mathcal C^\infty(V)$ are homomorphisms of algebras
  for every inclusion $V \subseteq U$ of open subsets of $M$, and that 
  for every open cover $(U_i)_{i \in I}$ of $M$ and functions
  $\phi_i \in \mathcal C^\infty(M)$ one has
  \[
    \left(\forall i, j \in I : \phi_i|U_i \cap U_j = \phi_j|U_i \cap U_j\right)
    \implies \exists! \phi \in \mathcal C^\infty(M) \, \forall i \in I :
    \phi|U_i = \phi_i.
  \]
\end{xca}

\begin{xca}
  Prove that a map $f\colon M \to N$ between manifolds is a morphism if and only if
  \[
    \forall \psi \in \mathcal C^\infty(M):
    f^{-1} \psi \in \mathcal C^\infty(N).
  \]
\end{xca}

\begin{xca}
  Prove that the composition $g \circ f$ or two morphisms $f\colon M \to N$
  and $g\colon N \to P$ between manifolds is again a morphism.
\end{xca}

\begin{xca}
  Prove that the inclusion morphism $i\colon U \to M$ for any open submanifold
  $U$ of a manifold $M$ is in fact a morphism.
\end{xca}

\begin{xca}
  Let $M$ and $N$ be two manifolds. Show that 
  \[
    \Set{(x \times y, U \times V): (x, U) \in \mathfrak U^\infty(M),
      (y, V) \in \mathfrak U^\infty(N)}.
  \]
  is an atlas of the cartesian product $M \times N$.
\end{xca}

\begin{xca}
  Show that the underlying topological space of the product $M \times N$ of
  two manifolds $M$ and $N$ coincides of the topological product of the topological 
  spaces underlying $M$ and $N$.
\end{xca}

\begin{xca}
  Prove that the projections $\pr_1\colon M \times N \to M$ and $\pr_2\colon
  M \times N \to N$ defined on a product manifold $M \times N$ are in fact
  morphisms.
\end{xca}