Question

If \(X\) and \(Y\) are topological spaces and \(y_{0} \in Y\), then \(X\) is homeomorphic to \(X \times\left\\{y_{0}\right\\}\) where the latter has the relative topology as a subset of \(X \times Y\).

Short answer

There is a homeomorphism between \( X \) and \( X \times \{y_0\} \) with the relative topology.

Step-by-step solution

Define the concept of homeomorphism

A homeomorphism is a bijective, continuous function with a continuous inverse between two topological spaces. This implies that the spaces are topologically equivalent.

Establish the map

Consider the map \( f: X \rightarrow X \times \{y_0\} \) defined by \( f(x) = (x, y_0) \).

Show that the map is bijective

For every \( x_1, x_2 \in X \), if \( f(x_1) = f(x_2) \), then \( (x_1, y_0) = (x_2, y_0) \), which means \( x_1 = x_2 \). Hence, \( f \) is injective. Given any \( (x, y_0) \in X \times \{y_0\} \), there exists an \( x \) such that \( f(x) = (x, y_0) \). Thus, \( f \) is surjective.

Verify that the map is continuous

In the subspace topology, a basic open set in \( X \times \{y_0\} \) is of the form \( U \times \{y_0\} \) where \( U \) is open in \( X \). The preimage of \( U \times \{y_0\} \) under \( f \) is \( U \), which is open in \( X \), proving that \( f \) is continuous.

Verify that the inverse map is continuous

The inverse map \( f^{-1}: X \times \{y_0\} \rightarrow X \) is given by \( f^{-1}(x, y_0) = x \). An open set in \( X \) is of the form \( U \) which translates to \( U \times \{y_0\} \) in \( X \times \{y_0\} \). Since \( f^{-1} \) takes \( U \times \{y_0\} \) to \( U \), it is continuous.

Conclude the proof

The function \( f: X \rightarrow X \times \{y_0\} \) is a bijective and continuous map with a continuous inverse, thus a homeomorphism.

Key concepts

Topological Spaces

Topological spaces are fundamental in understanding and studying the concept of homeomorphism. Think of a topological space as a set equipped with a collection of 'open sets.' This collection must contain the entire set itself and the empty set, and it must be closed under finite intersection and arbitrary union. This structure allows us to talk about continuity, convergence, and other important properties in a very general way.

In essence, topological spaces generalize and extend the idea of geometrical shapes, allowing mathematicians to rigorously study spaces in a broader sense. For instance:

• The real number line \(\mathbb{R}\) is a common example of a topological space.
• Another example is the set of all points on a plane, often referred to as the Euclidean plane.

Understanding topological spaces is crucial because it gives us the language and the tools to define and explore continuity and homeomorphic relationships.

Continuous Function

A continuous function is a bridge connecting domains in topological spaces. In simple terms, it ensures that small changes in input lead to small changes in output. Mathematically, a function \( f: X \to Y \) is continuous if, for every open set \( V \) in \( Y \), the preimage \( f^{-1}(V) \) is open in \( X \). This preservation of 'openness' is central to the concept.

This is crucial when considering homeomorphisms since both the function and its inverse need to be continuous. Imagine stretching or bending an object without tearing or gluing it. That's what continuity means: maintaining the 'essence' and properties of a space without disruption.

• For example, consider a continuous map from a circle to an ellipse. It stretches but doesn't break the circle.
• Another intuitive example could be a coffee mug and a donut, which are homeomorphic in topology due to their continuous shape transformations that preserve properties.

In topology, continuity isn't just about smoothness; it is about preserving the shape's fundamental properties.

Bijective Map

A bijective map is a function that perfectly pairs elements from one set to another. There are no leftovers or repeats. In mathematical terms, a function \( f: A \to B \) is bijective if it is both injective and surjective.

• An injective function (or one-to-one) means no two elements in \( A \) map to the same element in \( B \). This means \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).
• A surjective function (or onto) means every element in \( B \) is mapped to by some element in \( A \). This ensures the whole set \( B \) is covered.

Why is this important? For a homeomorphism, having a bijective map ensures there is a perfect match between the structures of the topological spaces involved. This perfect pairing guarantees that properties are preserved during transformation, as there are no missing or duplicated parts to suppress or exaggerate features of the space.
In topology, this means that there exists an inverse function that perfectly reverses the mapping, which is key for establishing equivalence between spaces.

Subspace Topology

Subspace topology is an important concept that helps in understanding spaces within larger contexts. Imagine taking a smaller piece of a bigger, more complex world. This smaller "subspace" inherits the properties of the larger space.

The subspace topology is defined for a subset \( Y \) of a topological space \( X \) with topology \( \tau \). The topology on \( Y \) consists of the intersections of \( Y \) with the open sets of \( X \). This way, every open set in the subspace is simply the intersection of the subset with some open set from the original space.

• This is helpful in proving homeomorphisms. When considering \( X \times \{y_0\} \) as a subspace of \( X \times Y \), the topology is simply inherited through intersections.
• It allows us to manipulate and analyze parts of a topological space without losing the essence of its environment.

Understanding subspace topology is crucial for effectively grasping relationships like homeomorphisms, showing how smaller parts of spaces interact coherently within larger structures.