Problem 15 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Show that ff \(f . X \rightarrow Y\) and \(g \cdot X \rightarrow Y\) are continuous maps from a topological space \(X\) into a Haucdorff space \(Y\) then the set of pomts \(A\) on whel these maps agrec, \(A=\\{x \in\) \(X \mid f(\mathrm{r})=g(x)\\}\), s closed. If \(A\) is a dense subset of \(X\) show that \(f=g\)

Step-by-Step Solution

Answer

The set \(A\) on which the maps \(f\) and \(g\) agree is a closed set and if \(A\) is a dense subset of \(X\), then \(f=g\).

Step 1: Understanding Properties of Continuous Maps and Hausdorff Spaces

A continuous map preserves the topological structure. A Hausdorff space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. The set \(A\) is defined as those points in \(X\) for which the maps \(f\) and \(g\) give the same result, i.e, \(A=\{x \in X | f(x)=g(x)\}\). In Hausdorff spaces, the property of closed sets aids in the proof as all singleton sets are closed.

Step 2: Proving \(A\) is closed

To show \(A\) is closed, it's equivalent to show that its complement in the space \(X\) is open. We consider a point in \(X-A\), so for this point \(f(x) \neq g(x)\). As \(Y\) is a Hausdorff space, there exist two disjoint open sets \(U\) and \(V\) in \(Y\) such that \(f(x) \in U\) and \(g(x) \in V\). Because \(f\) and \(g\) are continuous, \(f^{-1}(U)\) and \(g^{-1}(V)\) are open in \(X\). The intersection of these inverse images contains \(x\) and is contained in \(X-A\). So each point of \(X-A\) is an interior point, hence \(X-A\) is open in \(X\) and \(A\) is closed in \(X\).

Step 3: Proving that if \(A\) is dense then \(f=g\)

If \(A\) is dense in \(X\), then every point \(x\) in \(X\) is either a point in \(A\) or a limit point of \(A\). For a point in \(A\), \(f(x) = g(x)\) by the definition of \(A\). For a limit point \(x\) of \(A\), every neighbourhood of \(x\) in \(X\) contains a point of \(A\) other than \(x\). As \(f\) and \(g\) are continuous, \(f(x)\) and \(g(x)\) must be limit points of \(f(A)=g(A)\), hence \(f(x)=g(x)\). Thus for all points \(x\) in \(X\), \(f(x)=g(x)\) and hence \(f=g\).

Key Concepts

Continuous Maps

Continuous maps are an essential concept in topology. These maps ensure that the structure of a topological space is preserved during transformations. Imagine a function that "smoothly" translates each point in one space to another without tearing or creating jumps.

• A map is continuous if the inverse image of any open set in the target space is an open set in the starting space.
• This behavior keeps intact the 'closeness' of points in the original space.
• Continuous maps are crucial for maintaining the properties of shapes and spaces in mathematical analysis.

Continuous functions play a key role when dealing with various types of topological spaces, such as Hausdorff spaces. This ties directly into our next topic.

Dense Subset

A dense subset of a space is a subset that is as 'spread out' or 'full' as possible within that space.

• If you take any point in the space, a dense subset will have points arbitrarily close to it.
• In mathematical terms, a subset is dense in a space if every point of the space can be approximated as closely as desired by points from this subset.
• In practical terms, think of it like a fog that touches every nook and cranny of a landscape.

If set \(A\) is dense in \(X\), it ensures that the behavior or properties defined within \(A\) extend to the entire space \(X\). This concept is key when showing that certain properties or equations that hold in \(A\) must also apply throughout all of \(X\).

Topological Space

A topological space is a set that has a structure, defined by its open sets, allowing us to formalize the concept of convergence, continuity, and boundary.

• The open sets satisfy three main properties: the union of open sets is open, the finite intersection of open sets is open, and the space itself and the empty set are open.
• This structure is essential for discussing concepts like continuity and convergence.
• Almost every study concerning geometric or abstract objects begins by examining its topological space.

Understanding this concept allows us to formally define and work with ideas such as closed sets and continuous maps, bridging the gap between intuition and rigorous mathematics.

Closed Set

A closed set is the complement of an open set within a topological space. This means if a set is closed, its complement is open.

• Closed sets include all their boundary points; think of it as a box with all its sides shut.
• In Hausdorff spaces, every closed set can be separated by neighborhoods, which makes proving properties like those in the exercise easier.
• The intersection of any number of closed sets is closed, and the union of any finite number of closed sets is also closed.

In the context of the original problem, showing that a set where two continuous functions agree is closed helps ensure that their agreement isn't just coincidental but rather a structured property of the space.