Question

Let \(X\) and \(Y\) be topological spaccs and \(f . X \times Y \rightarrow X\) a continuous map. For each fixed \(a \in X\) show that the map \(f_{n}: Y \rightarrow X\) defined by \(f_{\theta}(v)=f(a, v)\) is contunous.

Short answer

Yes, with each fixed \(a \in X\), the map \(f_{\theta}: Y \rightarrow X\) defined by \(f_{\theta}(v)=f(a, v)\) is continuous.

Step-by-step solution

Define the function

The first step is to define the function \(f_{\theta}: Y \rightarrow X\), which is given by \(f_{\theta}(v) = f(a, v)\) where \(a\) is a fixed point in \(X\). By definition, for each \(a \in X\), we have a function \(f_{\theta}: Y \rightarrow X\).

Check the pre-image of an open subset

The next step is to check for any open subset \(U\) of \(X\), the pre-image under \(f_{\theta}\), i.e., \(f_{\theta}^{-1}(U)\) is open in \(Y\). Here \(f_{\theta}^{-1}(U) = \{v \in Y : f_{\theta}(v) \in U \}\).

Verify whether the pre-image is open

Observe that \(f_{\theta}^{-1}(U) = \{v \in Y : f_{\theta}(v) \in U \} = \{v \in Y : f(a, v) \in U \}\). This set is actually equal to \(f^{-1}(U \times Y)\), where \(f: X \times Y \rightarrow X\) is the original continuous map. Since \(f\) is continuous and \(U \times Y\) is open in \(X \times Y\), \(f^{-1}(U \times Y)\) is open in \(X \times Y\). Hence, \(f_{\theta}^{-1}(U)\) which is a subset of \(f^{-1}(U \times Y)\) is open in \(Y\). Hence \(f_{\theta}: Y \rightarrow X\) is continuous.

Key concepts

Continuous Functions

Continuous functions are a fundamental concept in topology, and they play a vital role in connecting different topological spaces.

When we talk about a function being continuous, we mean that small changes in the input lead to small changes in the output. This is similar to the intuitive idea of continuity in calculus, but in topology, the definition is framed using open sets.

For a function between two topological spaces to be considered continuous, the pre-image of every open set in the codomain must be an open set in the domain. In the original exercise, the function \(f_\theta: Y \rightarrow X\) is given, and we need to verify its continuity. We check that for any open set \(U\) in \(X\), the set \(f_\theta^{-1}(U) = \{v \in Y : f_\theta(v) \in U \}\) is open in \(Y\).

If this condition holds true for every open set \(U\) in \(X\), then the function \(f_\theta\) is continuous. This ensures that the mapping between spaces preserves the structure of open sets, maintaining the topological integrity across the transformation.

Topological Spaces

In topology, a topological space is a set equipped with a topology, a collection of open sets that satisfy certain axioms. Topological spaces are the central object of study in topology, much like how vector spaces are the core of linear algebra.

A topology on a set \(X\) is a collection of subsets of \(X\), which includes the empty set and \(X\) itself, such that any union of these subsets is also in the collection, and any finite intersection of these subsets is again in the collection. The elements of this collection are called open sets.

Understanding the concept of topological spaces helps us analyze and understand continuity and convergence in a generalized, more abstract sense. In the exercise, \(X\) and \(Y\) are both topological spaces, forming the domain and codomain for the continuous map \(f: X \times Y \rightarrow X\).

By considering the Cartesian product \(X \times Y\), we extend the ideas of topology to pairs of points, combining two distinct topologies. The continuity of a function defined on such products involves verifying the pre-images of open sets in the product topology remain open, underlining the complex but fascinating interplay between these mathematical spaces.

Open Sets

Open sets are a fundamental building block in topology, much akin to the role of intervals in calculus.

An open set is an essential concept as it helps define continuity and other fundamental ideas in topology. In any topological space, open sets are included in the topology of that space by definition. They need to satisfy criteria like being closed under arbitrary unions and finite intersections.

In context, a function is declared continuous if the pre-image of every open set in the codomain is an open set in the domain. For example, in our exercise, for a map \(f_\theta: Y \rightarrow X\), we verify continuity by determining if the pre-image of an open subset \(U\) of \(X\) is open in \(Y\).

• Every point in an open set can be indefinitely approached by other points within the set.
• The entire set does not include any boundary points, illustrating its non-closed nature.

Understanding open sets and their properties equips students with tools to grasp how spaces can transform under various mappings and to internalize why the concept of continuity relies heavily on the behavior of these sets in the function mapping process.