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General Relativity 01

Last updated: 2017-04-19 21:55:03 PDT.

### Topological space (definition dump)

**Definition (Topological space, open set)**: A topological space $(X,\mathcal J)$ is a set $X$ with a collection $\mathcal J$ of subsets of $X$ that satisfies:

- $X, \emptyset \in \mathcal J$;
- If $O_1, O_2, ... \subseteq \mathcal J$, then $\bigcup_\alpha\,O_\alpha \in \mathcal J$;
- If $n \in \mathbb Z^+, O_1, ... O_n \in \mathcal J$, then
$$ \bigcap_{i=1}^n O_i \in \mathcal J\,. $$

Sets in $\mathcal J$ are called "** open set**"s.

An example of a topology on $\mathbb R$ contains all the

open intervalsin $\mathbb R$. Thus the name "open set"s.

**Definition (Induced topology)**: If $(X,\mathcal J)$ is a topological space and $A$ is a subset of $X$, we may make $A$ into a topological space by defining the topology $\mathcal F = \{U\mid U=A\cap O, O\in\mathcal J\}$, then $(A, \mathcal F)$ forms a topology space. $\mathcal F$ is called *induced* (or *relative*) topology.

**Definition (Product topology)**: If $(X_1,\mathcal J_1)$ and $(X_2,\mathcal J_2)$ are both topological spaces, the direct prodct of both naturally forms a topological space $(X_1\times X_2, \mathcal J)$. $\mathcal J$ is called the **product** topology.

NOTEThis lifts the dimension of the topological space.

Open balls on $\mathbb R^n$ naturally form a topology.

**Definition (Continuous mapping)**: If $(X,\mathcal J)$ and $(Y,\mathcal K)$ are topological spaces, a map $f:X\to Y$ is ** continuous** if the inverse image $f^{-1}[O] \equiv \{x\in X \mid f(x) \in O\}$ maps every open set in $Y$ to an open set in $X$.

**Definition (Homeomorphism)**: If $f$ is continuous, one-to-one, onto, and its inverse is continues, then $f$ is called a ** homeomorphism**, and the spaces are said to be "

**".**

*homeomorphic*

NOTENot to be confused withhomomorphismandhomomorphic.

**Definition (Closed set)**: The complement of an open set is called a "closed set". Sets in a topology can be open, close, both, or neither.

**Definition (Connected)**: The topology is said to be conneted if the only subsets that are both open and closed are $X$ and $\emptyset$. $\mathbb R^n$ is connected.

**Definition (Closure)**: If $(X, \mathcal J)$ is a topological space, $\forall A \subseteq X$, the ** closure** $\overline A$ is the intersection of all open sets that contains $A$.

Properties:

- $\overline A$ is closed;
- $A \subseteq \overline A$;
- $\overline A = A \iff A$ is closed.

NOTEMeaning "to make a set closed". Closure of a set is unique and is necessarily in the topology.

**Definition (Interior, Boundary)**: Interior of $A$ is defined as the union of all the open sets contained in $A$. The boundy of $A$, denoted $\dot A$ (or $\partial A$), is defined as elements in $\overline A$ but not the interior of $A$, $\equiv \mathrm{int}(A)$.

NOTEalternatively, $\partial A \equiv \overline A \cap \overline {X \setminus A} $.

NOTEalternatively, $\mathrm{int}(A) = X \setminus \overline {X \setminus A} $.

NOTEalternatively, $\partial A \equiv \{ p \mid p \in X. O \in \mathcal J. p \in O \to (\exists \, a, b \in O. a \in A \wedge b \notin A) \}$.

**Definition (Hausdorff)**: A topological space is ** Hausdorff** if any two distinct points can be included in two disjoint open sets.

$\mathbb R^n$ is Hausdorff.

### Compactness

One of the most powerful notions in topology is that of compactness, which is defined as follows.

**Definition (Open cover)**: If $(X, \mathcal J)$ is a topological space and a collection of open sets $C=\{O_\alpha\}$ has $\bigcup_\alpha\,O_\alpha = X$, then $C$ is said to be an **open cover** of $X$, and $C$ "**covers**" $X$. Also if $Y$ is a subset of $X$, and $Y \subseteq \bigcup_\alpha\,O_\alpha$, then $C$ is said to be an **open cover** of $Y$ and $C$ "**covers**" $Y$. A subcollection of $C$ forms a subcover if it also covers $X$ (or $Y$).

**Definition (Compact space)**: If every open cover can be written as finite subcover, then the topological space is compact.

Alternative definitions of compact space. The following are equivalent:

- A topological space $X$ is compact.
- Every open cover a $X$ has a finite subcover.
- $X$ has a sub-base such that every cover of the space by members of the sub-base has a finite subcover (Alexander's sub-base theorem).
- Any collection of closed subsets of $X$ with the finite intersection property has nonempty intersection.
- Every net on X has a convergent subnet (see the article on nets for a proof).
- Every filter on X has a convergent refinement.
- Every ultrafilter on X converges to at least one point.
- Every infinite subset of X has a complete accumulation point.

**Definition (Open cover of a set, subcover of a set)**: If $(X, \mathcal J)$ is a topological space and $A$ is a subset of $X$. A open cover $U$ is a open cover of $A$ if $A \subset U$.
A subcover that also covers $A$ is called a subcover of $A$.

**Definition (Compact subset)**: $A$ is said to be compact if **every** open cover of A has a finite subcover.

The relation ship between compact space and compact subset is given by these two theorems:

- Compact subset of a Hausdorff space is closed.
- Closed subset of a compact space is compect.

**Heine-Borel Theorem**. A closed interval $[a, b]$ of $\mathbb R$ is compact.

Open interval $(0, 1)$ is not compact (since the open cover $O_\alpha = (1/\alpha, 1)$ has no finite subcover).

A subset of $\mathbb R$ is compact iff it is closed and bounded.

NOTEA unbounded set can totally be closed. For example, $\mathbb Z$ is obviously unbounded and closed.

Let $(X, \mathcal J)$ and $(Y, \mathcal K)$ be topological spaces. Suppose $(X, \mathcal J)$ is compact and $f: X \to Y$ is continuous. Then $f[X] \equiv \{y\in Y \mid y = f(x)\}$ is compact.

NOTEThis transfers compactness through homeomorphisms.

A continuous function from a compact topological space into $\mathbb R$ is bounded and attains its maximum and minimum values.

**Tychonoff theorem**: Product of compact topological spaces is compact. Given the axiom of choice, the number of such spaces can be infinite.

An application of these is that $S^n$ is compact, because 1) the sphere in $\mathbb R^{n+1}$ is closed and bounded, therefore compact; 2) there is a continuous function from $\mathbb R^{n+1}$ to $S^n$.

### Convergence of sequences

To extend the normal definition of sequence convergence, a sequence $\{x_n\}$ of points in a topological space $(X, \mathcal J)$ is said to converge to point $x$ if $\forall O \in \mathcal J .( x \in O \to \exists N \in \mathbb Z. \forall n > N. x_n \in O)$. $x$ is called the limit of the sequence.

A point $y \in X$ is said to be a **accumulation point** of $\{x_n\}$ if every open neiborhood of $y$ contains infinitely many points of the sequence.

NOTEThe difference between a limit and an accumulation point is that the former requires a particular set of infinite points in $\{x_n\}$. For example, the alternating sequence $1, -1, 1, -1, ...$ has two accumulation points $1$ and $-1$, but it does not have a limit.

**Definition (First countable)**: For every point $p$ in $X$, if there is a countable collection of open sets $\{O_\alpha\}$ that for every neiborhood $O$ of $p$, $O$ contains at least one element in $\{O_\alpha\}$.

**Definition (Second countable)**: There is a countable collection of open sets that every open set can be written as the union of some of the sets in the collections. The sets in that collection are called **basis**.

NOTEThe basis of alinear spaceis a collection ofvectors, s.t. every vector in the space is alinear combinationof the basis. The basis of atopological spaceis a collection ofopen sets, s.t. every opens set in the space is aunionof the basis.

NOTE$\mathbb R^n$ is second countable. Open balls with rational radii centered on rational coordinates can form a countable collection of open sets.

NOTEEvery second countable space is first countable.

The relationship between compactness and convergence of sequences is expressed by Bolzano-Weierstrass theorem:

**Bolzano-Weierstrass theorem** Let $(X, \mathcal J)$ be a topological space and let $A \subset X$

- If $A$ is compact, then every infinite sequence $\{x_n\}$ of points in $A$ has a accumulation point lying in $A$;
- Conversely, if $(X, \mathcal J)$ is second countable and every sequence in $A$ has an accumulation point in $A$, then $A$ is compact.

Thus, in particular, if $(X, \mathcal J)$ is second countable, $A$ is compact iff every sequence in $A$ has a convergent subsequence whose limit lies in $A$.

### Paracompactness

**Definition (Neighborhood)** Given $p$ in topological space $(X, \mathcal J)$, a ** neighborhood** $V$ of $p$ is a subset of $X$ that includes an open set $U$ containing $p$:

$$ V \subseteq X, \exists U \in \mathcal J, U \subseteq V, p \in U $$

NOTE: $V$ may not be open, but it contains a open set $U$ that contains $p$.

**Definition (Refinement of an open cover)**: Open cover $\{V_\beta\}$ of $X$ is said to be a refinement of open cover $\{O_\alpha\}$ of $X$ if $\forall V_\beta. \exists O_\alpha. V_\beta \subseteq O_\alpha$.

NOTERefinements forms a partially ordered set.

NOTESubcover is always a refinement of a open cover. A refinement of a open cover is not always a subcover.

**Definition (Locally finite)**: $\{V_\beta\}$ is **locally finite** if each $x \in X$ has an open neighborhood $W$ such that only finitely many $V_\beta$ satisfy $W \cap V_\beta \neq \emptyset$.

NOTECompactness requires a finite subcover, locally finiteness only requires a finte refinement. It is a weaker requirement.

**Definition (Paracompactness)**: A space is paracompact if every open cover has a locally finite refinement.

NOTE: Locally finiteness is weaker than finitenes of subcovers. Therefore every compact space is paracompact.

NOTE: (wiki) Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space. Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.

A paracomact manifold $M$ implies:

- $M$ admits a Riemannian metric and
- $M$ is second countable.

The most important implication is that a paracompact manifold $M$ will have a partition of unity.

**Definition (Partition of unity)**: If $(X, \mathcal T)$ is a topological space, and $R$ is a set of continuous functions from $X$ to unit interval $[0, 1]$, such that for every point $x \in X$:

- there is a neighborhood of $x$ where all but finite number of functions in $R$ are $0$, and
- the sum of all the functions at $x$ is $1$:
$$ \sum_{\rho\in R}\rho(x) = 1 $$

This is for the ease of defining integrals on the manifold.