2.2 Bases and Subspace Topologies
In the previous chapter we saw that in a metric space, any open set is a union of open balls, where in
general, infinitely many open balls are involved in the union. In a topological space it is convenient to
have some sub-collection of the open sets, called a basis, having the same property as the open balls in
a metric space.
Definition 2.2.1. Given a topological space \((X,\tau )\) a basis for \(\tau \) is a sub-collection \(B\subseteq \tau \) such that every
set in \(\tau \) is a union of sets from B.
Example 2.2.2. Let \(X=\{1,2,3\}\) and \(\tau =\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\},X,\emptyset \}\). Then \((X,\tau )\) is the discrete topological space. The collection \(B=\{\{1\},\{2\},\{3\},\emptyset \}\) is a basis
for \(\tau \) since: \(\emptyset =\emptyset \cup \emptyset \), \(X=\{1\}\cup \{2\}\cup \{3\}\), \(\{1\}=\{1\}\cup \emptyset \), \(\{2\}=\{2\}\cup \emptyset \), \(\{3\}=\{3\}\cup \emptyset \), \(\{1,2\}=\{1\}\cup \{2\}\), \(\{1,3\}=\{1\}\cup \{3\}\), \(\{2,3\}=\{2\}\cup \{3\},\{1,2,3\}=\{1\}\cup \{2\}\cup \{3\}\).
An equivalent formulation for the definition of a basis is the following: Given a set \(X\), a collection \(B\) of subsets of \(X\) is a basis for \(X\) if
- i
- \(X\) is a union of sets from \(B\).
- ii
- If \(B_1, B_2\in B\), then \(B_1\cap B_2\) is a union of sets from \(B\).
Theorem 2.2.3. Suppose that \(B\) is a basis for a non-empty set \(X\), and let
\(\tau =\{A\subseteq X: A \hspace {0.2cm} \text {is a union of sets from}\hspace {0.2cm} B\}\). Then \(\tau \) is a topology for \(X\).
Proof. By definition of \(B\), we have that \(X\in \tau \). In the statement of the theorem, it is understood that
we are allowed to take the union of no sets from \(B\), so \(\emptyset \in \tau \). If \(D=\bigcup _{i\in I} B_{i1}\), \(E=\bigcup _{j\in J}B_{j2}\) for some indexing sets \(I\) and \(J\), where
the \(B_{i1}\) and \(B_{j2}\) all belong to \(B\), then \(D\cap E=\bigcup _{(i,j)\in I\times J} B_{i1}\cap B_{j2}\) which is a union of sets from \(B\), hence a union of sets from \(B\).
Finally, as we have noted, a union of unions of sets from \(B\) is again a union of sets from \(B\).
□
Definition 2.2.4. A sub-basis for a topology \(\tau \) is a sub-collection \(B\subseteq \tau \) such that any set in \(\tau \) is a
union of finite intersections of sets from B.
A sub-basis tends to be smaller than a basis.
Example 2.2.5. In \(\mathbb {R}\), the collection of all open intervals is a basis for the usual topology. For
a sub-basis, it is enough to take intervals of the form \((-\infty ,b)\) and \((a,\infty )\). Since any open interval is either
one of these or else the intersection of two of them: \((a,b)=(-\infty ,b)\cap (a,\infty )\); and any open set in \(\mathbb {R}\) is a union of open
intervals.
Definition 2.2.6. Let X be a non-empty set and suppose that \(\tau _1\) and \(\tau _2\) are both topologies for X.
Then \(\tau _1\) is said to weaker than \(\tau _2\) if \(\tau _1 \subseteq \tau _2\), and then \(\tau _2\) is said to be stronger than \(\tau _1\).
Example 2.2.7. For the set \(X=\{a,b,c\}\), then in-discrete topology \(\tau _1=\{\{a,b,c\},\emptyset \}\) is weaker than the discrete topology \[\{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\},\emptyset \}=\tau _2\]
In fact, any topology for \(X\) is weaker than the discrete topology and the in-discrete topology is
weaker than any topology for\( X\).
Note. A set may be open in a topology for \(X\), but it may not be open in another topology for
\(X\). For example, for the set \(X=\{1,2,3\}\), \(A=\{2,3\}\) is an open set in the discrete topology \(\tau =\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\},\emptyset \}\) but is not open in the
topology \(\tau ^* =\{\{1,2\},\{1,2,3\},\emptyset \}\).
Just like for metric spaces, we may form subspaces for topological spaces.
Proposition 2.2.8. Let \((X,\tau )\) be a topological space and \(Y\) a subset of \(X\). Then
\(\tau _Y=\{A\cap Y:A\in \tau \}\) is a topology on \(Y\).
Proof. Since \(X,\emptyset \in \tau \), we have \(Y,\emptyset \in \tau _Y\).
Suppose that \(B_1,B_2,.........,B_n\) is a finite collection of sets in \(\tau _Y\). Then there exists \(A_1,A_2,...........,A_n\) in \(\tau \) such that
\[B_1=A_1\cap Y, B_2=A_2\cap Y,............,B_n=A_n\cap Y\]
Since \(A_1,A_2,........,A_n\) are in \(\tau \) and \(\tau \) is a topology, \(\bigcap ^n_{k=1}A_k\in \tau \). It follows that
\[\bigcap ^n_{k=1}B_k=\bigcap ^n_{k=1}(A_k\cap Y)=\Bigg (\bigcap ^n_{k=1}A_k\Bigg )\cap Y\in \tau _Y\]
Similarly, if \(\{B_{\lambda }:\lambda \in \Omega \}\) is any collection of sets in \(\tau _Y\), then \(\bigcup _{\lambda \in \Omega } B_{\lambda }\in \tau _Y\). Hence \(\tau _Y\) is a topology on \(Y\).
□
Definition 2.2.9. Given a topological space \((X,\tau )\) and a subset \(Y\subseteq X\), the topology
\(\tau _Y=\{A\cap Y:A\in \tau \}\) is called the subspace topology on \(Y\).
The subspace topology on \(Y\) is also called the relative or induced topology on \(Y\). We usually refer
to \(Y\) simply as a subspace of \(X\).
Example 2.2.10. Let \(X=\{\alpha ,\beta ,\gamma \}\) and let \(\tau =\{\{\alpha \},\{\beta \},\{\gamma \},\{\alpha ,\beta \},\{\alpha ,\gamma \},\{\beta ,\gamma \},\{\alpha ,\beta ,\gamma \},\emptyset \}\). Then \((X,\tau )\) is the discrete topological space. Now if \(Y=\{\alpha ,\beta \}\), then \begin {align*} \tau _Y &=\{A\cap Y: A\in \tau \}\\ &=\{\{\alpha \}\cap Y,\{\beta \}\cap Y,\{\gamma \}\cap Y,\{\alpha ,\beta \}\cap Y,\{\alpha ,\gamma \}\cap Y,\{\beta ,\gamma \}\cap Y,\{\alpha ,\beta ,\gamma \}\cap Y,\emptyset \cap Y\}\\ &=\{\{\alpha \},\{\beta \},\emptyset ,Y,\{\alpha \},\{\beta \},Y,\emptyset \}\\ &=\{\{\alpha \},\{\beta \},Y,\emptyset \} \end {align*}
is the topology induced by \(Y\). Hence \(\tau _Y\) is a subspace topology for \(\tau \) and \(Y\) is a subspace of \(X\).
Closed Sets
The notion of closed set in a metric space can be extended to a topological space. In this section we
will define and present some of the properties of closed sets in topological spaces.
Definition 2.2.11. Let \(X\) be a topological space. A subset \(F\) of \(X\) is said to be closed in \(X\) if its
complement \(X/F\) is open in \(X\).
Recall from De-Morgans theorem that the complement of the union of some collection of subsets
of some sets \(S\) is the intersection of the complement of the intersection of some collection of
subsets of \(S\) is the union of the complements of those sets. The following result therefore follows
directly from the definition of a topological space.
Proposition 2.2.12. Let \(X\) be a topological space. Then the collection of closed sets in \(X\) has the following properties
- i
- \(X\) and \(\emptyset \) are closed sets.
- ii
- the intersection of any collection of closed sets in \(X\) is itself a closed set in \(X\).
- iii
- the union of any finite collection of closed sets in \(X\) is itself a closed set in \(X\).
Proof. Each part is the complement of the corresponding axiom for open sets, via De Morgan’s laws.
(i)
\(X\) is closed because its complement \(\emptyset \) is open, and \(\emptyset \) is closed because its complement \(X\) is open.
(ii)
Let \(\{F_i\}\) be any collection of closed sets. Then \[X\setminus \bigcap _i F_i = \bigcup _i \left (X\setminus F_i\right ),\] a union of open sets, which is open. Hence \(\bigcap _i F_i\) is closed.
(iii)
For finitely many closed sets \(F_1,\dots ,F_n\), \[X\setminus \bigcup _{k=1}^{n}F_k = \bigcap _{k=1}^{n}\left (X\setminus F_k\right ),\] a finite intersection of open sets, which is open. Hence the union is closed. □
Note. The asymmetry is worth noticing and is not an accident of the proof. Arbitrary intersections of closed sets are closed but only finite unions are, exactly mirroring the axioms for open sets, where arbitrary unions and finite intersections are permitted. The standard witness is \[\bigcup _{n=1}^{\infty }\left [\frac 1n,\,1\right ] = (0,1],\] an infinite union of closed sets that is not closed.
Example 2.2.13. Let \(X=\{a,b,c,d,e\}\) and let \(\tau =\{X,\emptyset ,\{b\},\{b,c\},\{b,c,d\}\}\). Then \((X,\tau )\) is a topological space.
The open sets are \(\emptyset , X,\{b\},\{b,c\},\{b,c\}\) and \(\{b,c,d\}\). The closed sets are \(\emptyset ,X,\{a,c,d,e\},\{a,d,e\}\) and \(\{a,e\}\).
Note that:
- i
- In a topological space, a set may be both open and closed. In particular for any topological space \((X,\tau ),\emptyset \) and X are always both open and closed.
- ii
- On the usual topology \((\mathbb {R},\tau )\) the only sets that are both open and closed are \(\emptyset \) and \(\mathbb {R}\).
The open intervals \(a<x<b\) are open and the closed intervals \(a\leq x\leq b\) are closed.
The half-open intervals \(a<x\leq b\) and \(a\leq x<b\) are neither open nor closed
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