91Ó°ÊÓ

In differential geometry, a metric is a function that defines a distance between elements in a space. For a metric, certain properties must be satisfied:

• Non-negativity: The distance between two points is always non-negative: \( d(x, y) \geq 0 \).
• Identity of indiscernibles: The distance is zero if and only if the two points are the same: \( d(x, y) = 0 \iff x = y \).
• Symmetry: The distance between two points is the same in either direction: \( d(x, y) = d(y, x) \).
• Triangle Inequality: The direct distance between two points is less than or equal to the sum of the distances via a third point: \( d(x, z) \leq d(x, y) + d(y, z) \).

In the given exercise, we started by verifying that both forms of \( \bar{d} \), namely \( \bar{d} = d / (1 + d) \) and \( \bar{d} = \min(1, d) \), meet these metric properties. By affirming these, we can declare them as metrics, just like the original function \( d \). Understanding these properties is crucial for further grasping other concepts like open sets and homeomorphisms.

A homeomorphism is a special type of function between topological spaces, particularly those with metric properties. It ensures that two spaces are topologically the same, meaning there exists a continuous, bijective function with a continuous inverse function that maps one space onto the other. In other words, a homeomorphism maintains the structure of the space and translates it exactly.

For two metrics to be considered equivalent, the identity map \( 1:(X, d) \rightarrow (X, \bar{d}) \) must be a homeomorphism. This means that every open set in the original metric \( d \) corresponds to an open set in the new metric \( \bar{d} \). For our exercise, confirming this property for both \( \bar{d} = \frac{d}{1 + d} \) and \( \bar{d} = \min(1, d) \) assures us that they are indeed equivalent to the original metric \( d \). Essentially, a homeomorphism allows us to argue that the spaces defined by these metrics are the same in terms of topology.

In topology, an open set is a set that, informally speaking, does not include its boundary. Within any open set, you can 'move a little bit' without leaving the set. Open sets play a fundamental role in defining topological spaces and understanding how metrics relate to continuity.

To confirm that the identity map between \( d \) and \( \bar{d} \) is a homeomorphism, we need to show that the image of any open set in \( (X, d) \) remains open in \( (X, \bar{d}) \). For the specific cases in our exercise:

• For \( \bar{d} = \frac{d}{1 + d} \), the mapping preserves openness because the function \( \frac{d}{1 + d} \) is monotonically increasing and continuous.
• For \( \bar{d} = \min(1, d) \), any open set in \( d \)-metric space corresponds directly to an open set in the \( \bar{d} \)-metric space since it preserves the bounded distances without altering the openness property.

Thus, every open ball (a core example of an open set) remains open under these metrics. Understanding open sets is essential for grasping the concept of continuity and transformations between different metric spaces.

One key aspect of a metric is the triangle inequality. This property states that for any three points \( x, y, \text{and} z \) in the space, the direct distance from \( x \) to \( z \) is never greater than the sum of the distances from \( x \) to \( y \) and from \( y \) to \( z \): \( d(x, z) \leq d(x, y) + d(y, z) \). This property ensures that the shortest distance between two points is a straight path, aligning with our intuitive understanding of distance.

When examining both \( \bar{d} = \frac{d}{1 + d} \) and \( \bar{d} = \min(1, d) \), we must verify they adhere to this principle:

• For \( \bar{d}(x, z) = \frac{d(x,z)}{1 + d(x, z)} \), we use the fact that \( \frac{d(x, y)}{1 + d(x, y)} \) is a concave, non-increasing function, ensuring that the sum of transformed distances does not exceed \( \bar{d}(x,z) \).
• For \( \bar{d}(x, z) = \min(1, d(x,z)) \), the new metric directly reflects the strong triangle inequality property because \( \min(1, d(x, y) + d(y, z)) \) never exceeds \( \bar{d}(x, y) + \bar{d}(y, z)\).

Preserving the triangle inequality is vital for any function to be a valid metric, making it a cornerstone for validating new metrics.