Preliminaries from topology and analysis

Point-set topology

We will only be using the standard metric topology of $\mathbb{R}^n$ in this text, but its useful to learn somewhat more general definitions. A topology gives us a precise way to discuss continuity and closed and open sets on general spaces. Formally, if $S$ is a set and $T$ is a set of subsets of $S$ , then $T$ is a topology on $S$ if:

Both the empty set and X are elements of $T$
Any union of elements of $T$ is an element of $T$
Any intersection of finitely many elements of $T$ is an element of $T$

The pair $(S,T)$ (call it $S_T$ for brevity) is a topological space. The elements of a topology $T$ are the open sets of $S_T$ . A subset of $S_T$ is closed if its complement is open. A subset of $A$ can be open, closed, both, or neither. The empty set and $S$ itself are always both closed and open.

A subset $B$ of a topology $T$ is called a base of the topology $T$ if any set in $T$ can be formed from a union of sets in $B$ . In this case we can also say that $T$ is generated by $B$ . A simple example is that the standard topology of the real line is generated by the set of open intervals.

Perhaps the most useful non-Hausdorff topology is the Zariski topology, used in algebraic geometry. The base of the Zariski topology is the complement of any set of solutions to an algebraic equation (defined over a particular space such as $\mathbb{C}^n$ ).

The closure $\bar{A}$ of a subset $A \subset S$ is the set of all points $s$ in $S$ such that any open set containing $s$ also contains a point in $A$ . A subset $A$ is called dense in $S$ if $\bar{A} = S$ . A set is called nowhere dense if its closure does not contain an open set.

Standard metric topology on R^n

Any metric (distance function) $d$ on $\mathbb{R}^n$ induces a topology. A metric $d$ is a real-valued function of two variables that is symmetric ( $d(x,y) = d(y,x)$ ), non-negative ( $d(x,y) \ge 0$ ), only zero on the diagonal ( $d(x,y) = 0$ implies $y=x$ ), and satisfies the triangle inequality:

$d(x,y) + d(y,z) \ge d(x,z)$

A set with a metric is called a metric space.

We will generally stick to the Euclidean metric:

$d(x,y) = \left ( \sum_{i=1}^n (x_i - y_i)^2 \right )^{1/2}$

The open sets of a metric space are any union of open balls around any point $x$ (i.e. they are a base of the topology). An open ball of radius $\epsilon$ around $x$ is defined as the set

$B_{\epsilon}(x) = \{ y \ | \ d(x,y) < \epsilon \}$

In one dimension this is the open interval $(x-\epsilon, x + \epsilon)$ . In two dimensions it is the open disk of radius $\epsilon$ around $x$ , and in three dimensions it is the interior of the sphere of radius $\epsilon$ around $x$ .

Hausdorff Spaces

The definition of a topological space is very general - too general for some applications. Often, to preserve more of the structure of Euclidean space (and our intuition about its topology) it is useful to restrict attention to Hausdorff spaces (named after the mathematician Felix Hausdorff). A topological space $X$ is Hausdorff if any two distinct points of $X$ are contained in disjoint open sets. This is also called the $T_2$ property (the use of $T$ comes from Andrey Tychonoff, who studied a hierarchy of various separation axioms).

Covers and compactness

A collection of sets $C$ covers a topological space $X$ (possibly just a subset of a larger topological space) if the union of all of the sets in $C$ is equal to $X$ . If all of the sets in $C$ are open, it is called an open cover of $X$ .

A space for which every open cover has a countable subcover is called a Lindelöf space.

If every open cover of $X$ has a finite subcover, then $X$ is compact. If $X$ is contained in a metric space this is equivalent to $X$ being closed and bounded (the fact that these are equivalent is called the Heine-Borel theorem).

A space $X$ is locally compact if every point in $X$ has a compact neighborhood.

Limits

A point $y$ in a topological space is said to be a limit point of a sequence of points $x_1, x_2, \ldots$ if for every open set $S$ containing $y$ there is a number $N$ such that if $i > N$ , $x_i \in S$ . In a Hausdorff space, if a sequence has a limit point then it is unique.

Perfect sets

The set of all limit points $S'$ of a set $S$ is called the derived set of $S$ . If $S = S'$ , then it is called a perfect set. A perfect set is closed and contains no isolated points (a point $x \in S$ is isolated if there is a neighborhood of $x$ that does not contain any other points of $S$ ).

Connectedness

A set is connected if it cannot be represented as the union of two or more disjoint non-empty open subsets. This is one of the broadest notions of connectedness; for example path-connectedness is more restrictive. A set $S$ is path-connected if for any two points $x$ and $y$ in $S$ , there exists a continuous map $h$ of the interval $[0,1]$ into $S$ such that $h(0) = x$ and $h(1) = y$ .

A totally disconnected space is a set whose only connected subsets are singletons (single points).

The implicit function theorem

Here we quickly review one of the most important theorems from analysis, the implicit function theorem. We will state it for a one-dimensional family of scalar functions, but it generalizes to higher dimensions (even infinite dimensions under some assumptions):

The implicit function theorem: Let $P = P(\mu,x)$ be a differentiable function with $P(0,0) = 0$ and $\frac{\partial P}{\partial x}(0,0)$ invertible. Then there exist $\delta > 0$ and $\epsilon > 0$ such that $P(\mu, x) = 0$ has a unique solution with $|x| < \epsilon$ for each $|\mu| < \delta$ . This defines the function $x = g(\mu)$ . The function $g$ is differentiable and its derivative is $g' = -(\frac{\partial P}{\partial x})^{-1} \frac{\partial P}{\partial \mu}$ .

This theorem is proved in the text Dynamical Systems by Shlomo Sternberg (available free online).

Glossary

In addition to the glossary of this text, Wikipedia has a nice glossary of terms in topology.