2.1 Definition and Examples of Topological Spaces
Here we will define topological spaces and provide some examples of topological spaces. As we will see,
every metric space is a topological space.
Definition 2.1.1. Let \(X\) be a non-empty set. A collection \(\tau \) of subsets of \(X\) is called a topology on \(X\) if the following properties are satisfied:
- i
- \(X,\emptyset \in \tau \)
- ii
- \(\bigcap ^{n}_{k=1} A_k\in \tau \) whenever \(A_1,A_2,...........,A_k\in \tau \)
- iii
- \(\bigcup _{\lambda \in \Omega } A_k \in \tau \) whenever \(\{A_{\lambda }:\lambda \in \Omega \}\) is any collection of sets in \(\tau \)
The pair \((X,\tau )\) is called a topological space. The elements of \(X\) are called points and the sets in \(\tau \) are called
open sets.
Example 2.1.2. Let \((X,d)\) be a metric space and let \(\tau \) be the set of all open sets in \(X\) relative to the
metric \(d\). Then \((X,\tau )\) is a topological space.
Proof. Since both \(X\) and \(\emptyset \) are open sets in \((X,d)\), we have that \(X,\emptyset \in \tau \).
Now \(A_1,A_2,............,A_n\) be any finite collection of open sets in \(\tau \). Then \(\bigcap ^{n}_{k=1} A_k \in \tau \).
If \(\{A_{\lambda }:\lambda \in \Omega \}\) is any collection of sets in \(\tau \), then \(\bigcup _{\lambda \in \Omega } A_{\lambda }\) is an open set and so \(\bigcup _{\lambda \in \Omega } A_{\lambda } \in \tau \). Hence \((X,\tau )\) is a topological space.
A topological space arising from a metric space is said to be a metrizable topological space.
From the previous example, we see that any metric space gives rise to a topological space.
Since there can be several metrics defined on a set, there can also be several topologies defined
on a set.
The topology arising from the usual metric on \(\mathbb {R}\) is called the usual topology on \(\mathbb {R}\).
□
Example 2.1.3. Let X be any non-empty set and consider \(\tau =2^x\) to be the set of all subsets of X.
Then \((X,\tau )\) is a topological space.
Proof. Since both \(X\) and \(\emptyset \) are subsets of \(X\), we get that \(X,\emptyset \in \tau \).
Also if \(A_1,A_2,...........,A_n\) are subsets of \(X\), then \(\bigcap ^n_{k=1}A_k\) is a subset of X and so \(\bigcap ^n_{k=1}A_k \in \tau \).
For any collection \(\{A_{\lambda }:\lambda \in \Omega \}\) of subsets of \(X\), the set \(\bigcup _{\lambda \in \Omega } A_{\lambda }\) is a subset of \(X\), so that \(\bigcup _{\lambda \in \Omega } A_{\lambda } \in \tau \).
The topology \(\tau =2^x\) in this example is said to be the discrete topology on \(X\) and \((X,\tau )\) is called the discrete
topological space for X.
From the definition of the discrete metric on \(X\), we see that \(B(x,r)=\{x\}\) if \(r<1\) and \(B(x,r)=X\) if \(r\geq 1\). So that the open balls
with respect to the discrete metric on \(X\) are the singleton sets \(\{x\}\) and the whole set \(X\). Since open
balls in a metric space are open sets, and since the union of any collection of open sets is open,
this implies that every subset of \(X\) is an open set. Therefore the discrete metric on \(X\) induces the
discrete topology on X.
□
Example 2.1.4. Let \(X\) be any non-empty set and \(\tau =\{X,\emptyset \}\). Then \((X,\tau )\) is a topological space.
Proof. Obviously, \(X,\emptyset \in \tau \).
Also, since \(\tau =\{\emptyset ,X\}\), \(\bigcap ^n_{k=1}A_k\in \tau \) for any collection \(A_1,A_2,.........,A_n\) in \(\tau \) holds trivially.
Likewise, \(\bigcup _{\lambda \in \Omega }A_{\lambda } \in \tau \) for any collection \(\{A_{\lambda }:\lambda \in \Omega \}\) of sets in \(\tau \) holds trivially.
□
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.