Section Introduction
Many of the properties that we introduced in metric spaces (continuity, limit points, boundary) could be phrased in terms of the open sets in the space. With that in mind, we can broaden our concept of space by eliminating the metric and just defining the opens sets in the space. This produces what are called topological spaces.
Recall that the open sets in a metric space satisfied certain properties, including that the arbitrary union of open sets is open and any finite intersection of opens sets is open. We will now take these properties as our axioms in defining topological spaces.
Definition 12.1.
Let
\(X\) be a nonempty set. A set
\(\tau\) 6 of subsets of
\(X\) is said to be a
topology on
\(X\) if
\(X\) and \(\emptyset\) belong to \(\tau\text{,}\)
any union of sets in \(\tau\) is a set in \(\tau\text{,}\) and
any finite intersection of sets in \(\tau\) is a set in \(\tau\text{.}\)
A topological space is then any set on which a topology is defined. If \(X\) is the space and \(\tau\) a topology on \(X\text{,}\) we denote the topological space as \((X, \tau)\text{.}\) The elements of \(\tau\) are called the open sets in the topological space. When the topology is clear from the context, we simple refer to \(X\) as the topological space. Some examples are in order.
Preview Activity 12.1.
(a)
Suppose \(X = \{ a, b, c\}\text{.}\) Is the set \(\tau = \{a,b\}\) a topology on \(X\text{?}\) Justify your response.
(b)
Suppose \(X= \{a,b,c,d\}\text{.}\) Is the collection of subsets consisting of \(\tau = \{ \{a\}, \{b\}, \{a,b\} \}\) a topology on \(X\text{?}\) Justify your response. If not, what is the smallest collection of subsets of \(X\) that need to be added to \(\tau\) to make \(\tau\) a topology on \(X\text{?}\)
(c)
Suppose \(X= \{a,b,c,d\}\text{.}\) Is the collection of subsets consisting of
\begin{equation*}
\tau = \{\emptyset, \{a\}, \{b\}, \{d\}, \{a,b\}, \{a,d\}, X \}
\end{equation*}
a topology on \(X\text{?}\) Justify your response. If not, what is the smallest collection of subsets of \(X\) that need to be added to \(\tau\) to make \(\tau\) a topology on \(X\text{?.}\)
(d)
Suppose \(X= \{a,b,c,d\}\text{.}\) Is the collection of subsets consisting of
\begin{equation*}
\tau = \{\emptyset, \{a\}, \{b\}, \{c\}, \{d\}, \{a,b\}, \{a,c\}, \{a,d\}, \{b, d\}, \{c,d\}, X \}
\end{equation*}
a topology on \(X\text{?}\) Justify your response. If not, what is the smallest collection of subsets of \(X\) that need to be added to \(\tau\) to make \(\tau\) a topology on \(X\text{?}\)
(e)
Let \(F\) be the collection of finite subsets of \(\R\text{.}\) Let \(\tau = \{\emptyset, \R\} \cup F\text{.}\) First, list three members of \(F\) and three sets that are not in \(F\text{.}\) Next, is \(\tau\) a topology on \(\R\text{?}\) Justify your response.
(f)
Let \(\tau = \{\emptyset, \R, \{0\} \}\text{.}\) Is \(\tau\) a topology on \(\R\text{?}\) Justify your response.
(g)
Let \(X\) be a set and let \(\tau = \{\emptyset, X\}\text{.}\) Is \(\tau\) a topology on \(X\text{?}\) Justify your response.
(h)
Let \(X\) be a set and let \(\tau\) be the collection of all subsets of \(X\text{.}\) Is \(\tau\) a topology on \(X\text{?}\) Justify your response.
The symbol \(\tau\) is the Greek lowercase letter tau.
