Random Math Notes

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Xah Lee, 2007-10-15

Metric space.

The concept of distance is codified like this: .

A metric space is a tuple (M,d) where M is a set and d is a metric on M, that is, a function d:M×M→ℝ such that

d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
d(x, y) = d(y, x)     (symmetry)
d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality).

It follows d(x, y) ≥ 0     (non-negativity)
because
2d(x, y) = d(x, y) + d(y, x) ≥ d(x,x) = 0.

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Compact space.

A subset of Euclidean space R^n is called compact if it is bounded and closed.

Note: when a space is not bounded, it is necessarily not closed. Bounded just means they don't go beyond some point.

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Topological space.

The most common way to define a topological space is as a set X together with a collection T of subsets of X satisfying the following axioms:

The empty set and X are in T.

The union of any collection of sets in T is also in T.

The intersection of any pair of sets in T is also in T.

The collection T is called a topology on X, and the elements of X are called points. Under this definition, the sets in T are the open sets, and their complements in X are the closed sets. The requirement that the union of any collection of open sets be open is more stringent than simply requiring that all pairwise unions be open, as the former includes unions of infinite collections of sets.

By induction, the intersection of any finite collection of open sets is open. Thus, since the union of the empty collection is the empty set, and the intersection of the empty collection is (by convention) X, the definition above could also be stated using only the single axiom that T is closed under unions and finite intersections.

It's not clear to me how this definition is applied.

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Topology.

Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability, algebraic topology, which investigates such concepts as homotopy, homology, and geometric topology, which studies manifolds and their embeddings, including knot theory.