Topology

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\Z}{\mathbb{Z}}$

Topology¶

The obvious concrete example of a topology is the "usual" topology on $\R$, namely the set of sets of the form $(a,b) := \{y\in \R, y > a, y < b\}$, and all their (countable) unions.

Note the surprising fact that $(a,b)$ has no greatest point, while $[a,b] := \{y\in \R, y \geq a, y \leq b\}$, which is not open, does.

This notion of a toplogical structure on $\R$ is abstracted to an arbitrary topology on a set $S$, which is a set of subsets $U$ including $S$, $\{\}$, closed under countable unions and finite intersections.

The subsets $U$ are known as the open sets. Open sets allow us to talk about local properties, i.e. properties that may not hold true on the largest set of the topology (i.e. the whole space), but hold on some set of open sets which cover the space (i.e. whose union is the full space).

A topology is already enough algebraic structure to define the notions of a boundary, and compactness.

Boundaries¶

The interior $I(A)$ of a set $A$ is the largest open subset of $A$. The boundary is the complement of $I(A)$ in $A$. Have in mind the example of a disk and its boundary, a circle.

Continuity¶

Topological spaces and maps between them form a category TOP. The maps are an abstraction of continuous maps as seen on the reals.

There is the usual analytic definition of continuous maps on $\R$ involving a metric, and limits. The topological definition is much more abstract: a function $f$ such that $f^{-1}(S)$ is open.

One can see the relationship to the usual definition on $\R$ by considering a discontinuous function like the step function. Take the inverse of an open interval like $(0.5,2)$ and note that it is $[1, \infty]$, which is not open on $\R$ by the usual topology. So this function, which is discontinuous in the usual sense, is also discontinuous in the abstract sense.

Homeomorphisms¶

This is the word for isomorphisms in the category of topological spaces. So an equivalence between topological spaces.

Compactness¶

There's a definition of compactness for real numbers, which is that a compact set is closed and bounded. (Note that $[\alpha,\infty]$ or $[-\infty,\beta]$ or $[-\infty,\infty]$ is closed but not bounded, so closed and bounded mean different things.)

There's a topological definition that's much more abstract, that any open cover of a compact set has a finite subcover (this is equivalent for reals to the above definition by the Heine-Borel theorem.)

The important intuition - Terence Tao has a nice article on it - is that compactness is a property of infinite sets that are ``finite-ish''. Crucially, certain things that are true of functions on finite sets are true of continuous functions on compact sets.

Topological invariants¶

These are functions on the category of topological spaces $f$, such that $f(S_1)\neq f(S_2) \Rightarrow S_1 \neq S_2$. Compactness and connectedness are two examples.

The Euler characteristic of a topological space $S$ is Vertices(K)-Edges(K)+Faces(K), where $K$ is any polyhedron homeomorphic (i.e isomorphic) to $S$. Obviously this is a topological invariant.

Homology groups are another.

Manifolds¶

Manifolds (of dimension $n$) are topological spaces $M$ that are locally isomorphic to a ball in $\R^n$.

An important example of a space which is locally but not globally like $\R^n$ (in this case for $n=2$) is the surface of a ball, known as the 2-sphere.

Fiber bundles¶

It is not the case in all categories that $A/B \times B \cong A$, which is to say that we have spaces $E$ such that $E/G \cong M$ but not $E \cong M \times G$. Fiber bundles are such spaces $E$.

A fiber bundle is a topological space $E$ that is locally a product $B \times F$, in the sense that, locally, any projection $\pi$ to $B$ can be factored as $p_1 \circ \phi$, with $\phi : E \to B \times F$ an isomorphism, and $p_1 (x,y) = x$.

Think of a mobius strip as a simple example that is non-trivial (in the sense of not being globally a product space). $E$ is the whole space, $B$ is a circle$, and $F$ is a real interval.

Note that $E$ in the above example is a manifold itself (lying over $B$ which is also a manifold). This is generally true: fiber bundles are manifolds.

When a notion of a quotient is applicable, one can also view a fiber bundle $E$ as a map $E \to B := E/F$. This makes clear the issue: we want that $E/F = B$, but while this is certainly true for the trivial bundle $E\times F$, it is also true for a larger class of non-trivial bundles.

It is often useful to work "in coordinates", in the sense of specifying a fiber bundle by its local map $\phi_a : E \to B \times U_a$ and where two open sets overlap (i.e. $U = U_a \cap U_b$), a map $\Phi_{ab}$ such that $(\Phi_{ab} : U_a \cap U_b \times F \to U_a \cap U_b \times F) \circ (\phi_a : E \to U_a \cap U_b \times F) = (\phi_b : E \to U_a \cap U_b \times F)$. $E$ can then be recovered as a quotient $(\bigcup_{a\in I}U_a \times F)/ \sim $, where the equivalence relation is given by $\Phi_{ab}$.