Problem 8 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

If \(W, X\) and \(Y\) are topologieal spaces and the functions \(f: W \rightarrow X, g \quad X \rightarrow Y\) are toth contunuous, show that the function \(h=g \circ f, W \rightarrow Y\) is contimous.

Step-by-Step Solution

Answer

By the definition of continuity in topological spaces, we found that the pre-image of an open set in \( Y \) under the map \( h \), which equals to the pre-image of an open set in \( X \) under the map \( f \), is open in \( W \). This shows that the composition of two continuous functions in a topological space is also continuous.

Step 1: Define continuous functions in topological spaces

First, consider the definition of continuous maps in topology. A function \( f: W \rightarrow X \) is continuous if the pre-image of every open set in \( X \) is open in \( W \). That means for each open set \( U \subset X \), we have \( f^{-1}(U) \subset W \) is open.

Step 2: Express \( h \) in terms of \( f \) and \( g \)

Next, express the composite function \( h \) in terms of \( f \) and \( g \). We have \( h=g \circ f \), meaning \( h(w) = g(f(w)) \) for all \( w \) in \( W \). We are to prove that this function \( h \) is continuous.

Step 3: Use the definition of continuity to prove \( h \) is continuous

Now, take an open set \( V \) in \( Y \), we want to show that \( h^{-1}(V) \) is open in \( W \). By the continuity of \( g \), the set \( g^{-1}(V) \) is open in \( X \) . Because \( f \) is continuous, the set \( f^{-1}(g^{-1}(V)) \) is open in \( W \). Now, observe that \( h^{-1}(V) = f^{-1}(g^{-1}(V)) \). This implies that \( h^{-1}(V) \) is open in \( W \), confirming that the composition of \( f \) and \( g \), that is \( h \), is a continuous function.

Key Concepts

Continuous Functions

Continuous functions are a fundamental concept in topology, as well as many other areas of mathematics. To understand continuity in topological spaces, think of it as a bridge connecting the openness of sets between spaces. For a function \( f: W \rightarrow X \) to be continuous, a crucial requirement is that the pre-image of every open set in \( X \) must also be open in \( W \). This pre-image condition ensures that if you start with an open set in the codomain, when you "trace back" through the function to the domain, you end up with an open set there too.Imagine an open interval on the real line; a continuous function preserves this "openness" when mapping to a different space. This principle allows for the intuitive understanding of how functions "respect" the topology, or the structure of opens sets, in the spaces we are dealing with.
Here are key points about continuous functions in topological spaces:

• Preserve the open structure from codomain to domain
• Must satisfy: \( f^{-1}(U) \) is open in \( W \) for any open \( U \subset X \)

Topological Spaces

Topological spaces provide the broad stage where concepts like continuity are defined and understood. A topological space essentially consists of a set of points along with a notion of "neighborhoods" or "open sets." This structure is determined by a topology, which is a collection of subsets of the space that includes:

• The empty set and the full space itself
• Any union of these subsets
• Any finite intersection of these subsets

The role of these topological spaces is to provide a context where continuity and convergence can be generalized beyond numeric spaces. It allows mathematicians to study how spaces can be mapped into one another while preserving the notions of closeness and continuity. In this framework, functions that are continuous help maintain the structural elements, like open sets, making them vital for analysis in topology as well as complex systems like manifolds, spaces used in physics, and more.
This flexibility makes topological spaces a cornerstone concept in higher mathematics, allowing operations that rely on open set manipulations to be performed consistently no matter the specifics of the space.

Composite Functions

Composite functions involve creating a new function by combining two functions. For example, given two functions \( f: W \rightarrow X \) and \( g: X \rightarrow Y \), their composition is a new function \( h = g \circ f \) mapping \( W \) directly to \( Y \). Understanding composite functions is crucial when analyzing continuity across multiple spaces. This concept is sometimes tricky because it deals with how one sequence of functions maintains certain properties, like continuity, from start to finish.To show that the composite function \( h \) is continuous, it first requires that both \( f \) and \( g \) themselves are continuous. As the solution demonstrates, if you take an open set \( V \) in \( Y \), the pre-images flow back: \( g^{-1}(V) \) is open in \( X \) due to \( g \)'s continuity, and \( f^{-1}(g^{-1}(V)) \) is open in \( W \) due to \( f \)'s continuity. Thus, \( h^{-1}(V) = f^{-1}(g^{-1}(V)) \) is open in \( W \).
This connection offers:

• Ways to ensure continuity is preserved across mappings
• A method to "chain" continuous mappings into comprehensive transformations

As such, composite functions are extremely useful and fundamental in both theoretical contexts and applied mathematics, ensuring that ways spaces are related remain consistent and predictable.