Section 1.1 Topological spaces
The real numbers come equipped with a notion of nearness or closeness furnished by the absolute value function: namely, we define the distance \(d(a,b)\) between real numbers \(a,b\in\R\) as \(d(a,b)=\abs{a-b}\text{.}\) From this simple bit of structure springs all the fundamental concepts of real analysis: Cauchy sequences, limits, continuous functions, etc.
Topology can be seen as a vast generalization of this notion that abstracts away from the absolute value function, relying instead on the more abstract idea of open sets. Indeed, the notion of openness is introduced already in real analysis: a set \(U\subseteq\R\) is open if for all \(x\in U\) there is an \(\epsilon > 0\) such that if \(\abs{x-y} < \epsilon\text{,}\) then \(y\in U\text{.}\) Intuitively, you can think of this property as stating that \(U\) has a certain roomy or generous nature: if \(x\in U\text{,}\) and \(y\) is sufficiently close to \(x\text{,}\) then \(U\) has room for \(y\) too. Come on in!
We view the the three conditions of
Definition 1.1.1 as an attempt at axiomatizing this notion of openness (or roominess). This is the starting point of topology. The generality of this approach is part of topology’s great power. We can define a notion of open sets on any set
\(X\) whatsoever: i.e., we can choose a collection of subsets (called a
topology) that satisfies the axioms of
Definition 1.1.1. Furthermore, for a given set
\(X\) there are typically many distinct ways of making this choice; your topology on
\(X\) can be tailor-made to suit your particular needs. Beyond this generality and flexibility, the other key to topology’s effectiveness is its ability to transport to any topological space the concepts of continuity, limits, connectedness, and compactness familiar to us from real analysis. Once a set is given the structure of a topological space, these notions can be used as powerful tools for investigating its properties. It is for this reason that topology plays an important role in essentially all mathematical fields: analysis, geometry, algebraic geometry, number theory, etc.
Definition 1.1.1. Topological space.
A topological space is a pair \((X,\mathcal{T})\text{,}\) where \(X\) is a set, and \(\mathcal{T}\subseteq \mathcal{P}(X)\) is a collection of subsets of \(X\) satisfying the following axioms:
-
Trivial sets.
\(\emptyset\in \mathcal{T}\) and \(X\in \mathcal{T}\)
-
Closed under arbitrary unions.
Given any collection \(\{U_i\}_{i\in I}\) of elements \(U_i\in \mathcal{T}\text{,}\) we have \(U=\bigcup_{i\in I}U_i\in \mathcal{T}\)
-
Closed under finite intersections.
If \(U_1, U_2\in \mathcal{T}\text{,}\) then \(U_1\cap U_2\in \mathcal{T}\text{.}\)
In this case we call \(\mathcal{T}\) a topology on \(X\text{,}\) and an element \(U\in \mathcal{T}\) is called an open set of \(X\text{.}\) Although technically speaking a topological space is a pair \((X,\mathcal{T})\text{,}\) we often omit mention of the topology \(\mathcal{T}\) if there is no risk of confusion.
Topological specimen 1. Standard topology on \(\R\).
Let \(X=\R\) and define \(\mathcal{T}\) to be the collection of subsets \(U\subseteq \R\) satisfying the following property: for all \(x\in U\) there is an \(\epsilon > 0\) such that \((x-\epsilon, x+\epsilon)\subseteq U\text{.}\) The collection \(\mathcal{T}\) is a topology, called the standard topology on \(\R\text{.}\)
Proof.
The empty set satisfies the defining property of \(\mathcal{T}\) trivially. Thus \(\emptyset\in \mathcal{T}\text{.}\) For any \(x\in \R\text{,}\) we have \((x-1, x+1)\subseteq\R\text{.}\) Thus \(X=\R\in\mathcal{T}\text{.}\)
Assume \(U_i\in \mathcal{T}\) for all \(i\in I\text{,}\) and let \(U=\bigcup_{i\in I}U_i\text{.}\) We have
\begin{align*}
x\in U \amp \implies x\in U_i \text{ for some } i\in I\\
\amp \implies (x-\epsilon, x+\epsilon)\subseteq U_i \text{ for some } i\in I, \epsilon > 0\\
\amp \implies (x-\epsilon, x+\epsilon)\subseteq U \text{ for some } i\in I, \epsilon > 0 \text{.}
\end{align*}
This shows \(U\) satisfies the defining property of \(\mathcal{T}\text{,}\) and hence \(U\in \mathcal{T}\text{.}\)
Assume \(U_1, U_2\in \mathcal{T}\text{,}\) and let \(x\in U_1\cap U_2\text{.}\) Since \(x\in U_1\) and \(U_1\) is open, we have \((x-\epsilon_1, x+\epsilon_1)\subseteq U_1\) for some \(\epsilon_1 > 0\text{;}\) similarly, we have have \((x-\epsilon_2, x+\epsilon_2)\subseteq U_2\) for some \(\epsilon_2> 0\text{.}\) Letting \(\epsilon=\min\{\epsilon_1, \epsilon_2\}\text{,}\) we have \((x-\epsilon, e+\epsilon)\subseteq U_1\cap U_2\text{.}\) Thus \(U_1\cap U_2\in \mathcal{T}\text{,}\) as desired.
Topological specimen 2. Trivial and discrete topology.
Let \(X\) be a set.
The collection \(\mathcal{T}=\{\emptyset, X\}\) is a topology on \(X\text{,}\) called the trivial topology. This is the topology on \(X\) whose only open sets are the empty set and \(X\) itself.
The collection \(\mathcal{T}=\mathcal{P}(X)\) is a topology on \(X\text{,}\) called the discrete topology. This is the topology on \(X\) for which every subset of \(X\) is an open set.
Proof.
Let \(\mathcal{T}=\{\emptyset, X\}\text{.}\) By definition the given \(\mathcal{T}\) contains \(\emptyset\) and \(X\text{,}\) and thus axiom (1) is satisfied. Since any union or intersection involving \(\emptyset\) and \(X\) is equal to either \(\emptyset\) or \(X\text{,}\) it is clear that \(\mathcal{T}\) is closed under arbitrary unions and finite intersections. Thus axioms (2)-(3) are satisfied.
Let \(\mathcal{T}=\mathcal{P}(X)\text{.}\) By definition, \(\mathcal{T}\) is the set of all subsets of \(X\text{.}\) In particular, it contains \(\emptyset\) and \(X\text{,}\) and thus axioms (1) is satisfied. Furthermore, since any union or intersection of subsets of \(X\) is again a subset of \(X\text{,}\) we see that axioms (2)-(3) are satisfied.
Example 1.1.3. Some small topological spaces.
For each set
\(X\) describe all subsets
\(\mathcal{T}\subseteq \mathcal{P}(A)\) that satisfy the axioms of
Definition 1.1.1, and hence make
\((X,\mathcal{T})\) a topological space.
-
Empty set.
\(\displaystyle X=\emptyset\)
-
Singleton.
\(\displaystyle X=\{a\}\)
-
Doubleton.
\(\displaystyle X=\{a,7\}\)
Solution.
When
\(X=\emptyset\text{,}\) we have
\(\mathcal{P}(X)=\{\emptyset\}\text{.}\) Since by axiom (1) a topology
\(\mathcal{T}\) on
\(X\) must contain both
\(\emptyset\) and
\(X=\emptyset\text{,}\) we see the only option is
\(\mathcal{T}=\{\emptyset\}=\mathcal{P}(X)\text{.}\) We showed that
\(\mathcal{P}(X)\) is always a topology on
\(X\) in
Topological specimen 2. Thus
\(\mathcal{T}=\{\emptyset\}\) is the unique choice of topology for
\(X=\emptyset\text{.}\) (Note: in this case the trivial and discrete topologies are equal.)
When \(X=\{a\}\text{,}\) we have \(\mathcal{P}(X)=\{\emptyset, \{a\}\}\text{.}\) Since by axiom (1) a topology \(\mathcal{T}\) on \(X\) must contain both \(\emptyset\) and \(X=\{a\}\text{,}\) we see the only option is \(\mathcal{T}=\{\emptyset, \{a\}\}\text{,}\) which again is none other than the discrete topology. We conclude that \(\mathcal{T}=\{\emptyset, \{a\}\}\) is the unique topology on \(\{a\}\text{.}\) (Note: again in this case the trival and discrete topologies are equal.)
When \(X=\{a, 7\}\text{,}\) we have \(\mathcal{P}(X)=\{\emptyset, \{a\}, \{7\}, \{a,7\}\}\text{.}\) By axiom (1) a topology \(\mathcal{T}\) must contain \(\emptyset\) and \(X=\{a,7\}\text{.}\) It follows that there are exactly four possibilities for \(\mathcal{T}\) based on whether the sets \(\{a\}\) and \(\{7\}\) are elements of \(\mathcal{T}\text{:}\) i.e., the only possibilities are
\begin{align*}
\mathcal{T} \amp = \{\emptyset, \{a,7\}\} \\
\mathcal{T} \amp = \{\emptyset, \{a,7\}, \{a\}\} \\
\mathcal{T} \amp = \{\emptyset, \{a,7\}, \{7\}\} \\
\mathcal{T} \amp = \{\emptyset, \{a,7\},\{a\}, \{7\}\} \text{.}
\end{align*}
These are the four subsets of \(\mathcal{P}(A)\) that satisfy axiom (1). It is now straightforward to show that each of these choices also satisfies axioms (2) and (3). Thus there are exactly 4 distinct topologies we can define on \(X=\{a, 7\}\text{.}\)
Topological specimen 3. Cofinite topology.
Let \(X\) be a set. A subset \(U\subseteq X\) is cofinite in \(X\) if \(U^c=X-U\) is finite. The collection of subsets
\begin{equation*}
\mathcal{T}=\{\emptyset\}\cup \{U\subseteq X\colon U \text{ is cofinite}\}
\end{equation*}
is a topology, called the cofinite topology. In this topology a subset \(U\subseteq X\) is open if and only if it is either empty or cofinite.
Proof.
We treat each axiom separately.
We must show \(\emptyset, X\in \mathcal{T}\text{.}\) That \(\emptyset\in \mathcal{T}\) is specified explicitly in the definition of \(\mathcal{T}\text{.}\) Next since \(X^c=\emptyset\) is finite, the set \(X\) is cofinite in \(X\text{,}\) and hence an element of \(\mathcal{T}\text{.}\)
Let \(\{U_i\}_{i\in I}\) be any collection of elements of \(\mathcal{T}\text{:}\) i.e., for all \(i\in I\) we have \(U_i=\emptyset\) or \(U_i\) is cofinite in \(X\text{.}\) Let \(U=\bigcup_{i\in I}U_i\text{.}\) If \(U_i=\emptyset\) for all \(i\in I\text{,}\) then \(U=\emptyset\in \mathcal{T}\text{.}\) Otherwise there is an \(i_0\in I\) such that \(U_{i_0}\) is nonempty, hence cofinite. We have
\begin{align*}
U^c \amp = \left( \bigcup_{i\in I}U_i\right)^c \\
\amp = \bigcap_{i\in I} U_i^c \amp (\text{De Morgan's law})\\
\amp \subseteq U_{i_0}^c\text{.}
\end{align*}
Since \(U_{i_0}^c\) is finite, it follows that \(U^c\) is finite. It follows that \(U\) is cofinite, and hence an element of \(\mathcal{T}\) as desired.
Let \(U_1, U_2\in \mathcal{T}\text{,}\) and let \(U=U_1\cap U_2\text{.}\) We have \(U^c=U_1^c\cap U_2^c\text{.}\) Since \(U_i^c\) is either finite or equal to all of \(X\) for \(i=1,2\text{,}\) it is clear that \(U^c\) is either finite or equal to \(X\text{.}\) It follows that \(U\) is either cofinite or empty, and hence an element of \(\mathcal{T}\text{.}\)
Definition 1.1.4. Comparable topologies.
Two topologies \(\mathcal{T}, \mathcal{T}'\) on the set \(X\) are comparable if \(\mathcal{T}\subseteq \mathcal{T}'\) or \(\mathcal{T}'\subseteq\mathcal{T}\text{,}\) and incomparable otherwise. If \(\mathcal{T}\subseteq \mathcal{T}'\text{,}\) we say that \(\mathcal{T}\) is coarser than \(\mathcal{T}'\text{,}\) and that \(\mathcal{T}'\) is finer than \(\mathcal{T}\text{.}\) When \(\mathcal{T}\subsetneq\mathcal{T}'\) we say that \(\mathcal{T}\) is strictly coarser than \(\mathcal{T}'\) and \(\mathcal{T}'\) is strictly finer than \(\mathcal{T}\text{.}\)
Example 1.1.5. Comparing trivial, discrete, cofinite topologies.
Let \(X\) be a set, and let \(\mathcal{T}_{\operatorname{trivial}}, \mathcal{T}_{\operatorname{discrete}}, \mathcal{T}_{\operatorname{cofinite}}\) be the trivial, discrete, and cofinite topologies on \(X\text{,}\) respectively. Compare these topologies.
Solution.
First observe that in general the trivial topology \(\mathcal{T}_{\operatorname{trivial}}=\{\emptyset, X\}\) on \(X\) is coarser than any other topology, since by definition a topology must contain \(\emptyset\) and \(X\text{.}\) Similarly the discrete topology \(\mathcal{T}_{\operatorname{discrete}}=\mathcal{P}(X)\) is finer than any other topology, since by definition any topology is a subset of \(\mathcal{P}(X)\text{.}\) Thus we always have \(\mathcal{T}_{\operatorname{trivial}}\subseteq\mathcal{T}_{\operatorname{cofinite}}\subseteq\mathcal{T}_{\operatorname{discrete}}\text{.}\) The question naturally arises, whether and/or when these inclusions are strict. We consider a number of cases.
Case: \(\abs{X}\leq 1\).
In this case \(\mathcal{X}=\emptyset\) or \(\mathcal{X}=\{x\}\) (a singleton), in which case it is easy to see that \(\mathcal{T}_{\operatorname{trivial}}=\mathcal{T}_{\operatorname{cofinite}}=\mathcal{T}_{\operatorname{discrete}}=\{\emptyset, X\}
\text{.}\) (Note: when \(X=\emptyset\text{,}\) we have \(\{\emptyset, X\}=\{\emptyset\}=\{X\}\text{.}\))
Case: \(X\) finite, \(\abs{X}\geq 2\).
.
Since \(X\) is finite, it follows that all subsets of \(X\) are cofinite, and hence that \(\mathcal{T}_{\operatorname{cofinite}}=\mathcal{T}_{\operatorname{discrete}}=\mathcal{P}(X)\text{.}\) Since \(\abs{X}=n\geq 2\text{,}\) we have \(\abs{\mathcal{P}(X)}=2^n > 2=\abs{\mathcal{T}_{\operatorname{trivial}}}\text{.}\) It follows that in this case we have
\begin{equation*}
\mathcal{T}_{\operatorname{trivial}}\subsetneq \mathcal{T}_{\operatorname{cofinite}}=\mathcal{T}_{\operatorname{discrete}}=\mathcal{P}(X)\text{.}
\end{equation*}
Case: \(X\) infinite.
Let \(U\) be any finite nonempty subset of \(X\text{.}\) Since \(X\) is infinite and \(X=U\cup U^c\text{,}\) it follows that \(U^c\) must be infinite, and hence that \(U\) is not cofinite. We conclude that \(U\in \mathcal{T}_{\operatorname{discrete}}=\mathcal{P}(X)\) and \(U\notin \mathcal{T}_{\operatorname{cofinite}}\text{.}\) Thus \(\mathcal{T}_{\operatorname{cofinite}}\subsetneq \mathcal{T}_{\operatorname{discrete}}\) in this case. Next, fix in any element \(x\in X\) and let \(U=X-\{x\}\text{.}\) It is clear (a) that \(U\) is cofinite, and (b) that \(U\) is not equal to \(\emptyset\) or \(X\text{.}\) Thus \(U\in\mathcal{T}_{\operatorname{cofinite}}\) and \(U\notin\mathcal{T}_{\operatorname{trivial}}\text{,}\) proving that \(\mathcal{T}_{\operatorname{trivial}}\subsetneq \mathcal{T}_{\operatorname{cofinite}}\text{.}\) We conclude that
\begin{equation*}
\mathcal{T}_{\operatorname{trivial}}\subsetneq \mathcal{T}_{\operatorname{cofinite}}\subsetneq \mathcal{T}_{\operatorname{discrete}}
\end{equation*}
in this case.
