# Topology

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## 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 \(\Phi_{ab}\).