Topology/Lesson 1

What is a Topology?

The word "topology" has two meanings: it is both the name of a mathematical subject and the name of a mathematical structure. A topology on a set $X$ (as a mathematical structure) is a collection of what are called "open subsets" of $X$ satisfying certain relations about their intersections, unions and complements. In the basic sense, Topology (the subject) is the study of structures arising from or related to topologies.

Reading Assignment

The following reading is suggested to help supplement this lesson.

Wikibooks Topology book:
Wikipedia articles:

Definition (topology)

Let $X$ be a set. Then a topology on $X$ is a set $𝒯$ such that the following conditions hold.

${\emptyset, X} \subseteq 𝒯 \subseteq 2^{X}$ (where $2^{X}$ denotes the power set of X)
For $𝒮 \subseteq 𝒯$ we have $(⋃_{U \in 𝒮} U) \in 𝒯 .$
For finite sets $𝒮 \subseteq 𝒯$ we have $(⋂_{U \in 𝒮} U) \in 𝒯 .$

The set $X$ together with the topology $𝒯$ is called a topological space (or simply a space) and is commonly written as the pair $(X, 𝒯) .$ Or, when $𝒯$ is understood it may be omitted and we will simply say that $X$ is a topological space.

Examples

Here are some very simple examples of topological spaces. For these examples, $X$ can be any set.

Discrete topology: The collection $𝒯_{d} = 2^{X}$ is called the discrete topology on $X .$
Indiscrete topology: The collection $𝒯_{i} = {\emptyset, X}$ is called the indiscrete topology or trivial topology on $X .$
Particular point topology: Given a point $x_{0} \in X,$ the collection $𝒯_{x_{0}} = {U \subseteq X ∣ x_{0} \in U} \cup {\emptyset}$ is called the particular-point topology on $X .$

It is left as an exercise to verify that each of these three collections does indeed satisfy the axioms of a topology (conditions 1,2,3 in the definition above).

Reading supplement

Definition (open set, closed set,neighborhood)

Suppose that $(X, 𝒯)$ is a topological space.

Open set: A set $U \subseteq X$ is open if $U \in 𝒯 .$
Closed set: A set $A \subseteq X$ is closed if $A^{c} = (X ∖ A) \in 𝒯 .$
Neighborhood: For a point $x_{0} \in X$ a set $N \subseteq X$ is a neighborhood of $x_{0}$ if there is an open set $U \in 𝒯$ such that $x_{0} \in U \subseteq N .$

Definition (closed topology)

Alternate definition of a topology

Suppose that ${\emptyset, X} \subseteq 𝒮 \subseteq 2^{X} .$ Then $𝒮$ is a closed topology if

for any $ℛ \subseteq 𝒮$ we have $(⋂_{R \in ℛ} R) \in 𝒮$ and
for any finite collection $ℛ \subseteq 𝒮$ we have $(⋃_{R \in ℛ} R) \in 𝒮 .$

Show that for any set $X,$ the collection $𝒯$ is a topology on $X$ if and only if the collection $𝒮 = {T^{c} ∣ T \in 𝒯}$ is a closed topology on $X .$

Definition (interior, closure)

Let $(X, 𝒯)$ be a space and let $A \subseteq X .$

Interior: The interior of $A$ (denoted $int (A)$ ) is defined to be the union of all open sets contained in $A .$ In other words, $int (A) = ⋃_{\underset{U \in 𝒯}{U \subset A}} U .$
Closure: The closure of $A$ (denoted $\bar{A}$ ) is defined to be the intersection of all closed sets containing $A .$ That is, $\bar{A} = ⋂_{\underset{B^{c} \in 𝒯}{B \supset A}} B .$

Definition (basis)

Let $(X, 𝒯)$ be a space. Then a collection $ℬ \subseteq 𝒯$ is a basis if for any point $x_{0} \in X$ and any neighborhood $N$ of $x_{0}$ there is a basis element $B \in ℬ$ such that $x_{0} \in B \subseteq N .$

The benefit of talking about a basis is that sometimes describing every open set is unwieldy. For example, describing an open set in the Euclidean plane $ℝ^{2}$ would be difficult, but describing a basis is very easy. A basis of open sets in the plane is given by "open rectangles". That is $ℬ = {(a, b) \times (c, d) ∣ a < b, c < d \in ℝ}$ forms a basis.

Once a basis is determined, a set $U \subset X$ is open if it is the union of basis elements. That is, if $ℬ$ is a basis, then the topology is given by $𝒯 = {⋃_{B \in 𝒜} B ∣ 𝒜 \subseteq ℬ} .$

Definition (compact)

Let $(X, 𝒯)$ be a topological space. Then a set $K \subseteq X$ is compact if and only if every open cover of $K$ has a finite subcover.

Reading supplement

Lesson Exercises

Let $X$ be a three-point set. Then there are $2^{2^{3}} = 256$ different subsets of $2^{X} .$ How many of these are topologies on $X ?$ In other words, how many different 3-point topologies are there?
Can you find a formula for the number of topologies on an $n$ -point set?
Suppose that $ℬ \subseteq 2^{X}$ is such that for any $x \in X$ there is a set $B \in ℬ$ containing $x$ and that for any two sets $B_{1}, B_{2} \in ℬ$ such that $B_{1} \cap B_{2} \neq \emptyset$ there is a set $B_{3} \in ℬ$ such that $B_{3} \subseteq B_{1} \cap B_{2} .$ Show that the collection $𝒯 = {⋃_{A \in 𝒜} A ∣ 𝒜 \subseteq ℬ} \cup {\emptyset}$ is a topology on $X$ and that $ℬ$ is a basis for $𝒯 .$
Let $𝒮 \subseteq 2^{X}$ be such that for all $x \in X$ there is a set $S \in 𝒮$ which contains $x .$ Then show that the collection $ℬ = {⋂_{R \in ℛ} R ∣ ℛ \subseteq 𝒮 is finite}$ is a basis for a topology $𝒯$ on $X$ (using the criterion given in exercise 3). In this case, we call $𝒮$ a subbasis for $𝒯 .$
A basis $ℬ$ for a topology $𝒯$ is said to be minimal if any proper collection $𝒜 ⊊ ℬ$ is not a basis for $𝒯 .$ Given a set $X,$ find a minimal basis for the discrete topology $𝒯_{d} = 2^{X} .$
It is clear from the definition that $int (A) \subseteq A \subseteq \bar{A} .$ Show that if $A \subseteq B$ then $int (A) \subseteq int (B)$ and $\bar{A} \subseteq \bar{B} .$
Show that $int (int (A)) = int (A)$ and that $\overline{\bar{A}} = \bar{A} .$ Use these facts to show that $A \subseteq X$ is open if and only if $A = int (A)$ and is closed if and only if $A = \bar{A} .$
Is it true that for any set $A \subseteq X$ that $\overline{A^{c}} = (int (A))^{c} ?$ Give a proof or a counterexample.
Show that the collection $ℬ = {(a, b) ∣ a < b \in ℝ}$ ^[1] of open intervals is a basis for a topology on $ℝ .$ This is called the standard topology on $ℝ .$

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Notes

↑ where $(a, b) := {x \in ℝ : a < x < b}$

[1] where $(a, b) := {x \in ℝ : a < x < b}$

[1]