Definition 13.1.
A subset \(C\) of a topological space \(X\) is closed if its complement \(X \setminus C\) is open.
Section Introduction
We defined a closed set in a metric space to be the complement of an open set. Since a topology is defined in terms of open sets, we can make the same definition of closed set in a topological space. With the definition of closed set in hand, we can then ask if it is possible to define limit points, boundary, and closure in topological spaces and determine if there are corresponding properties for these ideas in topological spaces.
Preview Activity 13.1.
(a)
List all of the closed sets in the indicated topological space.
(i)
\((X, \tau)\) with \(X= \{a,b,c,d\}\) and \(\tau = \{\emptyset, \{a\}, \{b\}, \{a,b\}, X \}\text{.}\)
(ii)
\((X, \tau)\) with \(X= \{a,b,c,d,e,f\}\) and \(\tau = \{\emptyset,\{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e,f\}, X\}\text{.}\)
(iii)
\((X, \tau)\) with \(X = \R\) and \(\tau = \{\emptyset, \{0\}, \R\}\text{.}\)
(iv)
\((X, \tau)\) with \(X = \{a,b,c\}\) and \(\tau = \{\emptyset, \{a\}, \{b\},\{c\}, \{a,b\}, \{a,c\}, \{b,c\}, X \}\text{.}\) (What is the name of this topology?)
(v)
\((X, \tau)\) with \(X=\Z^+\) and \(\tau = \{\emptyset, X\}\) (this topology is called the indiscrete or trivial topology).
(b)
Using the examples from part (1), find (if possible), a set that is
(i)
both closed and open (if possible, find one that is not the entire set or the empty set)
(ii)
closed but not open
(iii)
open but not closed
(iv)
not open and not closed
(c)
In \(\R^n\) with the Euclidean metric, every single element set is closed. Does this property hold in the topological space \((X, \tau)\text{,}\) where \(X = \{a, b, c\}\) and \(\tau = \{\emptyset, \{a\}, \{a, b\}, \{a, c\}, X\}\text{?}\) Explain.
