Topology, Analysis, Statistics · Siblings

Topologies and σ-algebras are both collections of subsets that are closed under some operations. A topology is closed under arbitrary unions and finite intersections, and describes nearness. A σ-algebra is closed under complements and countable unions, and describes what can be measured.

Definitions

topology

High school

A topology on a set is a choice of which subsets count as “open.” It describes which points are near which, without requiring a definition of “distance”.

Undergraduate

Consider a set $X$ and a collection $\mathcal{T}$ of subsets of $X$. The collection $\mathcal{T}$ is a topology on $X$ if

  1. $\emptyset$ and $X$ are in $\mathcal{T}$,
  2. every union of sets in $\mathcal{T}$ is in $\mathcal{T}$, and
  3. every intersection of finitely many sets in $\mathcal{T}$ is in $\mathcal{T}$.

Each set in a topology is defined as an open set.

Examples
Non-examples
Properties

σ-algebra

Undergraduate

A σ-algebra is a collection of the subsets of a set that can be measured, such as the events whose probabilities are defined. If a set can be measured, then so can everything outside it, and so can every combination of countably many measurable sets.

Graduate

Consider a set $X$ and a collection $\mathcal{F}$ of subsets of $X$. The collection $\mathcal{F}$ is a σ-algebra on $X$ if

  1. $X$ is in $\mathcal{F}$,
  2. the complement of every set in $\mathcal{F}$ is in $\mathcal{F}$, and
  3. the union of every sequence of sets in $\mathcal{F}$ is in $\mathcal{F}$.
Examples
Non-examples
Properties
  • A σ-algebra also contains $\emptyset$ and is closed under countable intersections, by De Morgan’s laws.
  • The open sets of a topology generate a σ-algebra, the Borel σ-algebra: the smallest σ-algebra that contains every open set.

open set

Elementary (subsets of $\reals$)

For all real numbers $a$ and $b$ such that $a < b$, the interval of numbers strictly between $a$ and $b$ is called an open interval, and is written $(a, b)$.

High school (subsets of $\reals^n$)

Consider a set $U$ of points on the number line, in the plane, or in space. The set $U$ is open if every point of $U$ has a small ball around it that lies entirely inside $U$.

Undergraduate (metric spaces)

Consider a set $U$ in a metric space. The set $U$ is open if, for every point $x \in U$, there is an $\varepsilon > 0$ such that

\[B(x, \varepsilon) \subseteq U,\]

where $B(x, \varepsilon)$ is the open ball of radius $\varepsilon$ around $x$.

Graduate (topological spaces)

Consider a set $X$ with a topology $\mathcal{T}$. A set $U \subseteq X$ is open if $U \in \mathcal{T}$.

Examples
  • The empty set $\emptyset$ (definition)
    The empty set is open in $\reals$, because it has no points, so the condition on its points holds vacuously.
  • $(0, 1)$
    The interval $(0, 1)$ is open in $\reals$, because every point of $(0, 1)$ has a small interval around it inside $(0, 1)$.
  • The set of real numbers, $\reals$
    The set $\reals$ is open as a subset of itself, because every interval around a real number is inside $\reals$.
Non-examples
  • $[0, 1]$
    The interval $[0, 1]$ is not open in $\reals$, because every interval around $0$ contains negative numbers, which are not in $[0, 1]$.
  • $[0, 1)$, as a subset of $\reals$
    The interval $[0, 1)$ is not open in $\reals$, because every interval around $0$ contains negative numbers.
  • $\mathbb{Q}$, as a subset of $\reals$
    The set $\mathbb{Q}$ is not open in $\reals$, because every interval around a rational number contains irrational numbers.
  • The Cantor set
    The Cantor set is not open, because it contains no interval of positive length, so no point of the Cantor set (such as $0$) has a small interval around it inside the Cantor set.
Properties
  • Every union of open sets is open, and every intersection of finitely many open sets is open. An intersection of infinitely many open sets need not be open: the intersection of the intervals $(-1/n, 1/n)$ is $\setdef{0}$.
  • A set is open if and only if it equals its interior, and if and only if it contains none of its boundary points.

Missing definition: measurable_set

continuous

Elementary

Think of a function $y = f(x)$. The graph of $f(x)$ is a plot that shows the value of $y$ for each values of $x$ by plotting the value of $x$ on a horizontal axis and the value $y = f(x)$ on the vertical axis. The function is continuous if you can draw its graph without lifting your pencil.

Caveat. This only works when the domain is one unbroken interval. The function $x \mapsto 1/x$ is continuous, but its graph comes in two pieces because $0$ is not in its domain.
High school

Consider a function $f$. The function $f$ is continuous if

\[\lim_{x \to a} f(x) = f(a)\]

for every $a$ in the domain of $f$.

Undergraduate (metric spaces)

Consider a function $f : X \to Y$ between metric spaces. The function $f$ is continuous if for every $x \in X$ and every $\varepsilon > 0$, there exists $\delta > 0$ such that

\[d(f(x), f(y)) < \varepsilon \quad \text{for every } y \in X \text{ with } d(x, y) < \delta.\]

The choice of $\delta$ may depend on both $\varepsilon$ and $x$.

Graduate (topological spaces)

Consider a function $f : X \to Y$ between topological spaces. The function $f$ is continuous if the preimage $f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$.

Examples
Non-examples
Properties
  • Sums, products, and compositions of continuous functions are continuous.
  • Continuous images of compact sets are compact, and continuous images of connected sets are connected.
  • A continuous function on $[a, b]$ takes every value between $f(a)$ and $f(b)$ (the intermediate value theorem).

Missing definition: measurable_map

Examples

Example topology σ-algebra open set measurable_set continuous measurable_map Why
$\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$, on $X = \setdef{1, 2, 3}$ ✓✓––––
$\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$, on $X = \setdef{1, 2}$ ✓✗––––
All subsets of a set $X$ (its power set) ✓✓––––
The open sets of $\reals$ (the unions of open intervals) ✓✗––––
The Borel sets of $\reals$ ✗✓–––– The smallest σ-algebra on $\reals$ that contains every open interval.
The subsets of $\reals$ that are finite or have a finite complement ✗✗––––
Missing example: indicator_of_rationals_borel

Notes

See also

All distinctions · All definitions