Question

Let \(Z \subset Y \subset X, X\) a topological space. Give \(Y\) the induced topology. Show that the topology induced by \(Y\) on \(Z\) is the same as that induced by \(X\) on \(Z\).

Short answer

Answer: Yes, the topologies induced by Y on Z and X on Z are the same.

Step-by-step solution

Recall the definition of an induced topology

Given a topological space \((X, \tau)\) and a subset \(Y \subset X\), we define the induced topology on Y as \(\tau_Y = \{ Y \cap U : U \in \tau \}\). In simpler terms, it consists of the intersections of the open sets of the original topology with the subset Y. We can similarly define the induced topology on Z from Y, and from X.

Show that open sets in the topology induced by Y on Z have corresponding open sets in the topology induced by X on Z

Let \(W\) be an open set in the topology induced by Y on Z (denoted \(\tau_{ZY}\)). By definition of the induced topology, this means there exists an open set \(V\) in the topology induced by Y on X (denoted \(\tau_Y\)) such that \(W = Z \cap V\). Now, since \(V \in \tau_Y\), there must exist an open set \(U\) in the original topology \(\tau\) such that \(V = Y \cap U\). Therefore, we have \(W = Z \cap (Y \cap U) = (Z \cap Y) \cap U\). Since \(Z \subset Y\), we know that \(Z \cap Y = Z\), and therefore \(W = Z \cap U\). This shows that for every open set \(W\) in the topology \(\tau_{ZY}\), there exists an open set \(U\) in the original topology \(\tau\) such that \(W = Z \cap U\). This means that \(\tau_{ZY}\) is a subset of the topology induced by X on Z (denoted \(\tau_{ZX}\)).

Show that open sets in the topology induced by X on Z have corresponding open sets in the topology induced by Y on Z

Let \(W\) be an open set in the topology induced by X on Z (denoted \(\tau_{ZX}\)). By definition of the induced topology, this means there exists an open set \(U\) in the original topology \(\tau\) such that \(W = Z \cap U\). Since \(Y \subset X\) and \(Z \subset Y\), we know that \(Z \subset Y \cap U\). Also, since \(Y \cap U\) is an open set in \(\tau_Y\), we have \(W = Z \cap U = (Z \cap Y) \cap U \subset Y \cap U\). This shows that for every open set \(W\) in the topology \(\tau_{ZX}\), there exists an open set \(Y \cap U\) in the induced topology \(\tau_Y\) such that \(W \subset Y \cap U\). Thus, \(\tau_{ZX}\) is a subset of the topology induced by Y on Z (denoted \(\tau_{ZY}\)).

Conclude that the topologies are the same

In Steps 2 and 3, we showed that \(\tau_{ZY} \subseteq \tau_{ZX}\) and \(\tau_{ZX} \subseteq \tau_{ZY}\), which means that the two topologies are equal: \(\tau_{ZY} = \tau_{ZX}\). Thus, we have shown that the topology induced by Y on Z is the same as the topology induced by X on Z.

Key concepts

Topological Spaces

A topological space is a fundamental concept in the branch of mathematics called topology. It is essentially a set equipped with a collection of open sets that adhere to specific rules. These rules are:

• The whole set and the empty set must be included in the collection of open sets.
• Any union of open sets is also an open set.
• The intersection of any finite number of open sets is also an open set.

These requirements ensure that topological spaces provide a flexible and generalized framework for dealing with notions of convergence, continuity, and compactness in mathematics. The classic examples include familiar geometric shapes like lines and circles, but topological spaces can be far more abstract.

Open Sets

Open sets are crucial in understanding topology, as they define the structure of a topological space. In a given topological space \((X, \tau)\), the open sets are elements of the topology \(\tau\). They follow specific rules that allow topologists to talk about boundaries, limits, and interior points without needing to resort to a metric.Here are some important aspects of open sets:

• Any point within an open set has a neighborhood contained entirely within that set, which means there are no boundary points included.
• Open sets in the context of topology are generalized, meaning they do not necessarily adhere to everyday concepts of openness you encounter in geometry.
• The intersections and unions, specifically finite intersections of open sets are also open sets, adhering to the properties necessary for the defining structure of a topological space.

Subset Topology

When we have a subset \(Y\) of a topological space \(X\), we can still talk about open sets within \(Y\). The subset topology, also known as the induced topology, relies on the open sets of \(X\). To create the subset topology on \(Y\), we look at:

• The intersections of \(Y\) with open sets in \(X\). These intersections form open sets in the subset topology of \(Y\).
• This method allows \(Y\) to have a topological structure even though it is just a part of \(X\).

This ensures that the induced topology on \(Y\), denoted as \(\tau_Y\), maintains the original topological characteristics given by \(X\) while adapting them to the subset framework. Therefore, for any further subset \(Z \subset Y\), its subset topology derived directly from \(X\) or through \(Y\) ends up being the same.