Chapter 10: Problem 8

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.

Short Answer

Expert verified

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 by step solution

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.

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.

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.