Shall we begin?
Let us define a set $E$ as spacetime, with metric $g$ (All spacetimes have a Lorentzian metric, and so we shall endow $E$ with one.). Consider an infinitesimal $(n-1)$ dimensional sphere with radius $\epsilon$ and center $x\in E$, embedded in a spacelike $(n-1)$ dimensional slice of $E$. Let $\epsilon$ be small enough so that the metric at the point $x+\delta, |\delta|\leq|\epsilon|$, can be approximated by a Taylor expansion $g(x+\delta)=\sum_{k=0}\frac{\delta^k}{k!}\frac{\mathrm{d}^kg(x)}{\mathrm{d}x^k}$. Define this particular (non-unique) $(n-1)$ dimensional disk $D^{n-1}$ with radius $\epsilon$ as the \emph{Taylor neighborhood} $\mathcal{O}_x$ of the point $x$. This defines a topology on $E$ by $T=\{\mathcal{O}\subsetneq E|\forall x\in\mathcal{O}\exists\epsilon>0, \{y\in E|g(x,y)<\epsilon\}\subsetneq\mathcal{O}\}$. This provides a few local topological properties of spacetime:
- $\forall x,y\in E,\exists\mathcal{O}_x,\mathcal{O}_y\mbox{ such that }x\not\in\mathcal{O}_y$;
- $\forall x,y\in E,\exists\mathcal{O}_x,\mathcal{O}_y\mbox{ such that }\mathcal{O}_x\cap\mathcal{O}_y=\emptyset$;
- $\forall \mathcal{O}_x,\mathcal{O}_y,\exists\mbox{ closed neighborhoods }M,P\subset E\mbox{ such that }\mathcal{O}_x\cap\mathcal{O}_y=\emptyset$.
Consider the set $S$ of all possible sequences in $E$. Let $l$ be some finite number. Then, a bounded sequence is defined as a converging sequence $(x_n)$: $\forall U\in\{\mbox{set of all (Taylor) neighborhoods of $x$}\}=\mathcal{C}(x)\exists N\in\mathbb{N}\mbox{ such that }\forall n\geq N\Rightarrow x_n\in U$. We will consider $E$ to have the property that all sequences are complete, i.e., the sequence converges to some $l\in E$. Write $S_C$ as the set of all unbounded sequences factored out from $E$. Let $E_C$ be the set that results from this factorization. $E_C$ is the compactification of $E$. Let us define $\partial E$ as the set of all points in $E$ such that for some $P\in E$, the neighborhood of $P$ contains points which lie in $E$ and also do not lie in $E$.
We would now like to clarify what was meant by ``results from this factorization'' in the previous paragraph. A sequence is an ordered open set. The union of sequences is also a sequence, and the same hold for intersections. There also exists the ``empty'' sequence, and the sequence containing all the points of $E\backslash\partial E$ (we say $E\backslash\partial E$ and not $E$ because sequences are open). Therefore, the set of all sequences is a topology. In fact, by grouping together elements of $E$ to form ordered sequences, it is seen that the topology generated by $S$ is the ``ordered'' discrete topology.
We now attempt to find if $S_C$ is a topology. Consider two disjoint sequences in $S_C$. These sequences are complete and bounded. Therefore, there exists some finite $l\in E$ to which the sequences converge. (The $l$ for each sequence may be different.) The union of these sequences is also a sequence, and the intersection is the empty sequence. The sequence containing all the points of $E_C$ is complete if $E_C$ is compact. By the construction of $E_C$, $E_C$ is compact, and therefore since complete subspaces (if these subspaces are open, they are sequences) of $E_C$ are closed (since the compactification of a metric space is a metric space), and sequences as defined above are open, complete sequences in $E_C$ do not exist, with the exception of the empty sequence and $E_C\backslash\partial E_C$ itself. $S_C$ is then just the set containing the empty sequence and $E_C\backslash\partial E_C$, i.e., $S_C$ provides the trivial topology on $E_C\backslash\partial E_C$.
This is interesting... What does this mean for the separability of $E$ and $E_C$? We'll find out next time! So long, friends!
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