Question

If \(\operatorname{card}(X) \geq 2\), there is a topology on \(X\) that is \(T_{0}\) but not \(T_{1}\).

Short answer

Yes, the topology \(\{ \emptyset, \{a\}, X \}\) is \(T_0\), not \(T_1\).

Step-by-step solution

Understanding the Problem

The problem asks us to prove that there exists a topology on a set \(X\) such that the topology is \(T_0\) but not \(T_1\), given that the cardinality of \(X\) is at least 2. A topology is called \(T_0\) if for any two distinct points, there is an open set containing one of the points but not the other. It is called \(T_1\) if every single point is contained in an open set that excludes any other point.

Choosing the Set X

Since we know the cardinality of \(X\) is at least 2, let's consider the simplest case where \(X = \{a, b\}\). This satisfies the condition \(\operatorname{card}(X) \geq 2\).

Defining a Topology on X

We need to define a topology on the set \(X = \{a, b\}\). Consider the topology \(\mathcal{T} = \{ \emptyset, \{a\}, X \}\). Here, \(X\) and the empty set \(\emptyset\) are always in the topology by definition of topology, and now we add \(\{a\}\) as an open set.

Checking the T0 Property

For the topology \(\mathcal{T} = \{ \emptyset, \{a\}, X \}\), check the \(T_0\) separation: for \(a\) and \(b\), the open set \(\{a\}\) contains \(a\) but not \(b\). There is no open set containing \(b\) but not \(a\), however, the presence of one such open set (\(\{a\}\)) for the pair \(\{a, b\}\) is enough to satisfy the \(T_0\) condition.

Checking the T1 Property

Check for the \(T_1\) property: a topology is \(T_1\) if for any pair of distinct points \(a\) and \(b\), there are open sets containing \(a\) and \(b\) separately such that \(a\) is not in the open set containing \(b\) and vice versa. Here, \(\{b\}\) does not exist as an open set, violating the \(T_1\) condition.

Conclusion

The topology \(\mathcal{T} = \{ \emptyset, \{a\}, X \}\) on \(X = \{a, b\}\) is \(T_0\) because it distinguishes between \(a\) and \(b\) using the open set \(\{a\}\), but it is not \(T_1\) because \(\{b\}\) is not an open set. This proves the desired result.

Key concepts

T0 Space

A topology is called a \( T_0 \) space if for any two distinct points in the space, there exists an open set containing one of the points, but not the other. This means if you pick two different points, you should be able to tell them apart using open sets. It doesn't necessarily separate them completely like in a more strict topological space, which makes \( T_0 \) spaces less restrictive.
For example, in the topology \( \mathcal{T} = \{ \emptyset, \{a\}, X \} \) on the set \( X = \{a, b\} \), the open set \( \{a\} \) contains point \( a \) but excludes point \( b \). This satisfies the \( T_0 \) condition because you can distinguish between \( a \) and \( b \) by the open set \( \{a\} \).
This simple distinction property is what characterizes \( T_0 \) spaces, which can be essential in certain mathematical contexts where distinguishing elements is important, but a full separation is not required.

T1 Space

In contrast to \( T_0 \) spaces, \( T_1 \) spaces take the idea of separation a step further. A topology is a \( T_1 \) space if for any two distinct points, there are open sets that include each point and exclude the other points. This means two points must be completely separate by their respective open sets.
For the previous topology \( \mathcal{T} = \{ \emptyset, \{a\}, X \} \), it is not a \( T_1 \) topology because there is no open set like \( \{b\} \) that contains only \( b \) and not \( a \). To be \( T_1 \), both \( \{a\} \) and \( \{b\} \) would need to be open sets, allowing you to distinguish them completely with unique open sets.
So when you pick points \( a \) and \( b \), you must find two separate open sets. This level of separation is useful when you need definite distinctions between all points in a topology.

Open Sets

Understanding open sets is crucial in topology, as they form the building blocks of the space structure. An open set in a topology can be thought of as a "container" that satisfies certain properties such as union and intersection. The rules for building a topology include:

• The whole set \( X \) and the empty set \( \emptyset \) must be included in the collection of open sets.
• The union of any number of open sets must also be an open set.
• The intersection of a finite number of open sets must also be an open set.

In the context of the example with \( X = \{a, b\} \), the open sets chosen were \( \{\emptyset, \{a\}, X\} \). This satisfies the criteria mentioned; however, notice the lack of complete point separation in \( \mathcal{T} \).
Each topology you define on a set affects the properties like being \( T_0 \) or \( T_1 \), demonstrating the importance of how open sets are chosen.