Topology vs. σ-algebra
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
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”.
Consider a set $X$ and a collection $\mathcal{T}$ of subsets of $X$. The collection $\mathcal{T}$ is a topology on $X$ if
- $\emptyset$ and $X$ are in $\mathcal{T}$,
- every union of sets in $\mathcal{T}$ is in $\mathcal{T}$, and
- 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
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The open sets of $\reals$ (the unions of open intervals)
The open sets of $\reals$ form a topology because unions and finite intersections of open sets are open.
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$\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$, on $X = \setdef{1, 2, 3}$
The collection $\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$ is a topology on $\setdef{1, 2, 3}$, because it contains $\emptyset$ and $\setdef{1, 2, 3}$, and every union and intersection of its sets is in it.
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All subsets of a set $X$ (its power set)
The power set of $X$ is a topology on $X$: the discrete topology, in which every set is open.
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$\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$, on $X = \setdef{1, 2}$
The collection $\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$ is a topology on $\setdef{1, 2}$, because it contains $\emptyset$ and $\setdef{1, 2}$ and is closed under unions and intersections.
Non-examples
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The Borel sets of $\reals$
Every set is a union of one-point sets, which are Borel, but not every set is Borel. So the Borel sets are not closed under arbitrary unions.
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The subsets of $\reals$ that are finite or have a finite complement
The finite and cofinite subsets of $\reals$ do not form a topology, because they are not closed under arbitrary unions: the union of the one-point sets $\setdef{n}$, for $n \in \nats$, is $\nats$, which is neither finite nor cofinite.
Properties
- Every metric defines a topology, whose open sets are the unions of open balls.
- The complement of an open set need not be open. The complements of open sets are called closed.
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.
Examples
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$\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$, on $X = \setdef{1, 2, 3}$
The collection $\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$ is a σ-algebra on $\setdef{1, 2, 3}$, because it contains the complement of each of its sets, and every union of them.
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All subsets of a set $X$ (its power set)
The power set of $X$ is a σ-algebra on $X$, because it contains every subset of $X$, so it is closed under every set operation.
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The Borel sets of $\reals$
The Borel sets of $\reals$ form a σ-algebra: the smallest σ-algebra that contains the open sets.
Non-examples
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The open sets of $\reals$ (the unions of open intervals)
The complement of $(0, 1)$ is not open.
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$\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$, on $X = \setdef{1, 2}$
The collection $\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$ is not a σ-algebra on $\setdef{1, 2}$, because it does not contain $\setdef{2}$, the complement of $\setdef{1}$.
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The subsets of $\reals$ that are finite or have a finite complement
The finite and cofinite subsets of $\reals$ do not form a σ-algebra. They are closed under complements and finite unions, but not countable unions: the union of the one-point sets $\setdef{n}$, for $n \in \nats$, is $\nats$, which is neither finite nor cofinite.
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.
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 $(a, b)$.
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 $U$.
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 $x$.
Consider a set $X$ with a topology $\mathcal{T}$. A set $U \subseteq X$ is open if $U \in \mathcal{T}$.
Examples
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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.
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$(0, 1)$
The interval $(0, 1)$ is open in $\reals$, because every point of $(0, 1)$ has a small interval around it inside $(0, 1)$.
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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
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$[0, 1]$
The interval $[0, 1]$ is not open in $\reals$, because every interval around $0$ contains negative numbers, which are not in $[0, 1]$.
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$[0, 1)$, as a subset of $\reals$
The interval $[0, 1)$ is not open in $\reals$, because every interval around $0$ contains negative numbers.
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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
Missing definition: continuous
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
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The axioms differ in two ways: a topology allows any union (even uncountable) but not complements, while a σ-algebra allows complements but only countable unions.
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Continuous maps pull open sets back to open sets; measurable maps pull measurable sets back to measurable sets.