Problem 11 - Analysis Now

Question

(Topological direct sum.) Let \(\left(X_{1}, \tau_{1}\right)\) and \(\left(X_{2}, \tau_{2}\right)\) be topological spaces and let \(X\) denote the disjoint union of \(X_{1}\) and \(X_{2} .\) Find the topology \(\tau\) on \(X\) that contains \(X_{1}\) and \(X_{2}\) and for which the relative topology on \(X_{j}\) is \(\tau_{j}\) for \(j=1\), 2. Show that \(\tau\) is the final topology corresponding to the embedding maps \(l_{j}: X_{j} \rightarrow X\) for \(j=1,2\).

Step-by-Step Solution

Answer

\(\tau\) is defined such that \(U \subseteq X\) is open if \(l_j^{-1}(U) \in \tau_j\). This is the final topology for embeddings \(l_j: X_j \rightarrow X\).

Step 1: Understanding the Problem

We need to find a topology \(\tau\) on the disjoint union \(X\) of the topological spaces \((X_1, \tau_1)\) and \((X_2, \tau_2)\). This topology should ensure that the relative topology on \(X_j\) is exactly \(\tau_j\) for \(j=1,2\). Moreover, we need to demonstrate that this topology \(\tau\) is the final topology for the embedding maps \(l_j: X_j \rightarrow X\).

Step 2: Constructing the Disjoint Union

The disjoint union \(X\) is the set \(X = X_1 \sqcup X_2\), consisting of elements of \(X_1\) and \(X_2\) but considered as different from one another, even if they are Equal. The maps \(l_1: X_1 \rightarrow X\) and \(l_2: X_2 \rightarrow X\) simply attach all points of \(X_1\) and \(X_2\) onto \(X\).

Step 3: Defining the Topology \(\tau\)

The topology \(\tau\) on \(X\) is defined such that a set \(U\) is open in \(\tau\) if its intersection with each \(X_j\) is open in \(\tau_j\). Formally, \(U \subseteq X\) is in \(\tau\) if \(l_j^{-1}(U) \in \tau_j\) for \(j=1,2\). This ensures that \(\tau_1\) and \(\tau_2\) are the relative topologies on \(X\).

Step 4: Showing \(\tau\) is the Final Topology

In order for \(\tau\) to be the final topology for the maps \(l_1\) and \(l_2\), for any topological space \((Y,\sigma)\) and any functions \(f_1: X_1 \rightarrow Y\) and \(f_2: X_2 \rightarrow Y\), there should exist a unique function \(f: X \rightarrow Y\) such that \(f \circ l_j = f_j\) for \(j=1,2\), and \(f\) is continuous if and only if each \(f_j\) is continuous. Since an open set \(U\) in \(X\) satisfies that \(l_j^{-1}(U) \in \tau_j\), continuity of \(f\) is guaranteed precisely when \(f_j = f \circ l_j\) is continuous, proving \(\tau\) is indeed the final topology.

Key Concepts

Disjoint Union

When we talk about the disjoint union in topology, we're usually referring to a construction that allows us to combine two or more topological spaces into one larger space. Imagine having two completely separate rooms, one for * Space 1: \( X_1 \)* Space 2: \( X_2 \)We place these rooms together as part of a big building, which represents space \( X \). Even though the rooms are next to each other, the contents inside one room don't mix with the other.

Similarly, in the disjoint union of topological spaces \( X = X_1 \sqcup X_2 \), the elements from \( X_1 \) remain distinct from those in \( X_2 \), even if they are numerically or descriptively equivalent. This ensures we maintain the unique identity of each element, treating them as separate entities. This separation is crucial in keeping their respective topological properties intact.

Final Topology

The concept of a final topology is tied to how we create a new topology that encompasses multiple spaces. It's like forming a bigger neighborhood from smaller ones while keeping their characteristics.

In the problem's context, the final topology \( \tau \) on the disjoint union \( X \) arises from the embedding maps \( l_1: X_1 \rightarrow X \) and \( l_2: X_2 \rightarrow X \). These maps "embed" the individual spaces \( X_1 \) and \( X_2 \) into the larger space \( X \).
* To achieve this, a set \( U \subseteq X \) is considered open in \( \tau \) if, when snapped back to each small space, it remains open. This keeps the inherent structure of each smaller space intact.* The final topology also provides a universal property: any function from the disjoint union to another topological space can be broken down into functions from each separate space while retaining continuity.This property is akin to solving a big puzzle using smaller connected pieces, ensuring a cohesive fit all around.

Relative Topology

Imagine having a big park \( X \) with specific picnic sections \( X_1 \) and \( X_2 \). These sections can have their own rules or customs, which we refer to as the relative topology.

In our mathematical context, each \( X_j \) (where \( j = 1, 2 \)) within space \( X \) has a relative topology \( \tau_j \). This is derived from the larger topological space but tweaked to fit each section solely. It’s like taking general park rules and adapting them for picnic areas. * Essentially, a set is open in the relative topology \( \tau_j \) if it can be expressed as the intersection between an open set in the primary space \( X \) and \( X_j \) itself.
By focusing on these intersections, the relative topology guarantees that each section maintains its identity, safeguarding its distinct rules as part of the overall structure.

Embedding Maps

Embedding maps are like special delivery lines connecting smaller spaces to a bigger one. They allow each smaller space \( X_1 \) and \( X_2 \) to "embed" or establish their existence within the larger space \( X \).

These maps, \( l_1: X_1 \rightarrow X \) and \( l_2: X_2 \rightarrow X \), act like IDs, ensuring every point from \( X_1 \) and \( X_2 \) appears uniquely within \( X \). This is crucial for distinguishing identical points in different sections of the disjoint union.
* When an element from \( X_1 \) moves to \( X \) via \( l_1 \), it retains its identity, special to that original space. The same process applies to \( X_2 \) with \( l_2 \).* Embedding maps are fundamental in constructing the final topology as they define how each point in the disjoint spaces makes its way into a single, cohesive structure of \( X \). These maps ensure that every bit of the smaller spaces is included and accurately represented in the larger perspective.