Topology Notes

By Xah Lee. Date: .

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.

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.


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 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.


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.

Complete metric space

Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

Intuitively, a space is complete if there are no “points missing” from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. sqrt[2] is “missing” from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to “fill all the holes”, leading to the completion of a given space, as explained below.

Dense set

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation).

Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space the empty set and the one-point sets are connected; in a totally disconnected space these are the only connected subsets.

An important example of a totally disconnected space is the Cantor set. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers.

A topological space X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X are the one-point sets.

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