Introduction
In previous chapters, we analyzed specific behaviors of metric spaces such as
convergence,
boundedness,
continuity,
completeness,
connectedness, and
compactness. However, we have not yet defined when two different
spaces are "essentially" the same.
When data passes through the layers of a neural network, we often want the "intrinsic structure" of the data manifold
to be preserved. While isometry (preserving exact distances) is often too rigid for modern data analysis, we need a
more flexible concept of equivalence known as topological equivalence. It describes a scenario where one space can be
continuously deformed into another without tearing or gluing. This concept ensures that even if a latent space is stretched or twisted,
the fundamental connectivity of the data remains intact.
Topologically Equivalent Metrics
Before comparing entirely different spaces, we first examine how different distance functions on the same underlying
set compare to one another. In data analysis, the exact distance values might be distorted by transformations or scaling.
Understanding when two metrics yield the exact same collection of open sets - and therefore preserve the exact same convergent
sequences and continuous functions - is crucial when selecting an appropriate metric representation for a dataset without
altering its intrinsic topological structure.
Definition: Comparability of Metrics
Suppose \(X\) is a set and \(d\) and \(e\) are metrics on \(X\). Then,
-
\(d\) is topologically stronger than \(e\) and \(e\) is topologically weaker than
\(d\) if and only if every open subset of \((X, e)\) is open in \((X, d)\).
-
\(d\) and \(e\) are topologically equivalent if and only if \(d\) is both weaker and
stronger than \(e\).
-
\(d\) and \(e\) are not comparable if and only if \(d\) is neither topologically stronger
nor topologically weaker than \(e\).
There are multiple ways to describe the Comparability of metrics.
Theorem: Criteria for Topological Strength of Metrics
Suppose \(X\) is a set and \(d\) and \(e\) are metrics on \(X\). The following statements are logically
equivalent.
-
Every open ball of \((X, e)\) includes an open ball of \((X, d)\) with the same center.
(The "same center" requirement is essential: topological strength is a local property at each
point, so what must be compared are neighbourhoods of the same point in both metrics.)
-
Every open subset of \((X, e)\) is open in \((X, d)\).
-
Every closed subset of \((X, e)\) is closed in \((X, d)\).
-
The identity function from \((X, d)\) to \((X, e)\) is continuous.
-
Every sequence that converges in \((X, d)\) converges in \((X, e)\) to the same limit.
- For every metric space \(Z\) and every function \(g: X \to Z\): if \(g\) is continuous with
respect to \(e\), then \(g\) is continuous with respect to \(d\) (where the metric on \(Z\)
is the same in both statements).
-
For every metric space \(Z\) and every function \(g: Z \to X\): if \(g\) is continuous with
respect to \(d\), then \(g\) is continuous with respect to \(e\) (where the metric on \(Z\)
is the same in both statements).
Proof of (2) ⇔ (4): Continuity of the Identity Function
Let us show the fundamental equivalence between the topological strength of metrics and the
continuity of the identity function (Criterion 2 and 4): \(d\) is topologically stronger than \(e\) if and only
if the identity function \(f:(X, d) \to (X, e)\) is continuous.
Suppose \(d\) is topologically stronger than \(e\). By definition, every open subset \(U\) in \((X, e)\) is also
open in \((X, d)\). For the identity function \(f\), the preimage of \(U\) is \(f^{-1}(U) = U\).
Since \(U\) is open in \((X, d)\), the
topological characterization of continuity
(preimage of any open set in the codomain is open in the domain) gives that \(f\) is continuous.
Suppose \(f: (X, d) \to (X, e)\) is continuous. Let \(U\) be an open subset in \((X, e)\). By the
topological characterization of continuity,
the preimage \(f^{-1}(U)\) must be open in \((X, d)\). Since \(f^{-1}(U) = U\), \(U\) is open in \((X, d)\).
Therefore, \(d\) is topologically stronger than \(e\).
This straightforward proof highlights why "comparing topologies" is fundamentally identical to analyzing the continuity of
mappings between them.
Proof Sketch: Remaining Equivalences
(1) ⇔ (2). (1) is the "ball version" of (2). Assuming (1), any open set \(U\) in \((X, e)\)
and any \(x \in U\) admit an \(e\)-ball \(B_e(x; r) \subseteq U\); by (1), this contains a \(d\)-ball
\(B_d(x; s)\), so \(x\) is an interior point of \(U\) in \((X, d)\), proving \(U\) is \(d\)-open.
Conversely (2) ⇒ (1): every \(e\)-ball \(B_e(x; r)\) is \(e\)-open, hence \(d\)-open, hence contains
a \(d\)-ball centred at \(x\).
(2) ⇔ (3). Open and closed subsets are set-theoretic complements:
\(F\) is closed iff \(X \setminus F\) is open. Hence the family of open sets in a topology determines,
and is determined by, the family of closed sets. The equivalence follows directly.
(4) ⇔ (5). Identity-map continuity is equivalent to its
sequential version,
so (4) says precisely that \(d\)-convergent sequences are \(e\)-convergent to the same limit.
(4) ⇔ (6), (4) ⇔ (7). Any map \(g: X \to Z\) being \(e\)-continuous can be read as
\(g \circ \text{id}_{d \to e}\) being continuous; composing with the identity \(\text{id}_{d \to e}\) —
continuous by (4) — transfers continuity from \(e\) to \(d\). The dual statement handles maps
\(g: Z \to X\). So (4) is what lets the identity translate continuity hypotheses between the two
metrics on \(X\). \(\square\)
This establishes the criteria for topological strength; by applying this result symmetrically to both \(d\) and \(e\), we
obtain the criteria for topological equivalence as stated in the following corollary.
Corollary: Criteria for Topological Equivalence
Two metrics \(d\) and \(e\) on a set \(X\) are topologically equivalent if and only if:
-
The collection of open (closed) subsets of \((X, d)\) is the same as that of \((X, e)\).
-
The identity functions from \((X, d)\) to \((X, e)\) and from \((X, e)\) to \((X, d)\) are both continuous.
-
Every convergent sequence of \((X, d)\) is convergent in \((X, e)\) with the same limit, and vice versa.
-
Every function from \(X\) into a metric space is continuous with respect to \(d\) if and only if
it is continuous with respect to \(e\) assuming the metric on the codomain is fixed.
-
Every function from a metric space into \(X\) is continuous with respect to \(d\) if and only if
it is continuous with respect to \(e\) assuming the metric on the domain is fixed.
Proof:
Each criterion in the corollary is obtained by conjoining the corresponding criterion of the preceding theorem
with its reversal (swap the roles of \(d\) and \(e\)). Since topological equivalence means \(d\) is both stronger
and weaker than \(e\), each item reduces to a symmetric pair of already-proved statements. \(\square\)
In Summary, when we can say \(d\) and \(e\) are equivalent metrics on \(X\), \((X, d)\) and \((X, e)\) hold the same
open (closed) subsets,
dense subsets,
compact subsets,
locally compact subsets,
connected subsets,
convergent sequences, and
continuous functions
with \(X\) as domain (codomain).
Uniformly Equivalent Metrics
While topological equivalence preserves basic continuity and limit points, it fails to preserve "global" metric properties such as
Cauchy sequences and completeness. To guarantee that a space remains complete, or that an algorithm's convergence properties are not
destroyed when swapping metrics, we must introduce a stricter condition that bounds the variation globally: uniform equivalence.
When we can say \(d\) and \(e\) are uniformly equivalent metrics on \(X\), besides topologically equivalent metrics condition,
\((X,d)\) and \((X, e)\) hold the same
Cauchy sequences,
totally bounded subsets, and
uniformly continuous functions with \(X\)
as domain (codomain).
Lipschitz Equivalent Metrics
The strongest form of metric equivalence requires that the distances under one metric are bounded by a constant multiple of the distances under another.
This condition guarantees that the geometric distortion between the two metrics is strictly linearly limited. In machine learning, Lipschitz equivalence
is heavily relied upon in optimization theories, bounding gradients, and ensuring the theoretical stability of neural networks.
Definition: Lipschitz Equivalence of Metrics
Suppose \(X\) is a set and \(d\) and \(e\) are metrics on \(X\). We say that \(d\) is Lipschitz stronger
than \(e\) and that \(e\) is Lipschitz weaker than \(d\) if and only if the identity function
from \((X, d)\) to \((X, e)\) is a Lipschitz function. Moreover, \(d\) and \(e\) are Lipschitz equivalent
if and only if each of \(d\) and \(e\) is Lipschitz stronger than the other.
When we can say \(d\) and \(e\) are Lipschitz equivalent metrics on \(X\), besides uniformly equivalent metrics condition, \((X,d)\) and \((X, e)\)
hold the same
bounded subsets,
subsets with the nearest-point property, and
Lipschitz functions with \(X\) as domain (codomain).
Equivalent Metric Spaces
Having established how to compare different metrics on a single set, we now elevate these concepts to compare entirely distinct
metric spaces. This brings us to the formal definition of a homeomorphism - a mapping that establishes a perfect
topological correspondence between two spaces. This continuous, invertible correspondence between spaces forms the foundational base
for defining manifolds, which is the core mathematical structure behind Geometric Deep Learning.
Definition: Homeomorphism (Topologically Equivalent Metric Spaces)
Suppose \((X, d)\) and \((Y, e)\) are metric spaces.
Then \(X\) and \(Y\) are said to be homeomorphic or topologically equivalent if and
only if there exists a bijection \(f: X \to Y\) that is continuous and has continuous inverse. Such a function \(f\)
is called a homeomorphism.
Definition: Lipschitz Equivalent Metric Spaces
Suppose \((X, d)\) and \((Y, e)\) are metric spaces.
Then \(X\) and \(Y\) are said to be Lipschitz equivalent if and only if there exists a bijection
\(f: X \to Y\) that is Lipschitz continuous and whose inverse is also Lipschitz continuous.
These three notions form a strict hierarchy: every Lipschitz equivalence is uniform, and every uniform
equivalence is a homeomorphism, but neither converse holds (as Example 2 below will demonstrate).
Finally, let us observe two fundamental examples that bridge theoretical analysis and practical applications,
leading us toward the study of manifolds.
Example 1: Equivalence of Norms in \(\mathbb{R}^n\) (Lipschitz Equivalence)
In \(\mathbb{R}^n\), consider the \(L_1\) (Manhattan), \(L_2\) (Euclidean), and \(L_\infty\) (Chebyshev) norms.
It is a fundamental theorem that all norms on a finite-dimensional vector space are equivalent. Specifically,
they are Lipschitz equivalent. For any \(x \in \mathbb{R}^n\), we have the standard inequalities:
\[
\|x\|_\infty \le \|x\|_2 \le \|x\|_1 \le n\|x\|_\infty.
\]
Because these metrics bound each other up to a constant factor, the identity mappings between them are Lipschitz continuous.
For instance, \(\|x - y\|_\infty \le \|x - y\|_1\) shows that the identity \((\mathbb{R}^n, \|\cdot\|_1) \to (\mathbb{R}^n, \|\cdot\|_\infty)\)
is Lipschitz with constant \(1\), while \(\|x - y\|_1 \le n \|x - y\|_\infty\) gives the reverse direction with constant \(n\).
In machine learning, this guarantees that whether you use \(L_1\) or \(L_2\) regularization, the underlying topological properties
(such as convergence of sequences) remain strictly identical, even though the geometric shape of the unit ball changes.
Example 2: \((-1, 1)\) and \(\mathbb{R}\) (Topological Equivalence)
Consider the open interval \(X = (-1, 1)\) and \(Y = \mathbb{R}\) equipped with the standard Euclidean metric.
Let \(f: (-1, 1) \to \mathbb{R}\) be defined by:
\[
f(x) = \tan\left(\frac{\pi}{2}x\right).
\]
This function is a bijection, and both \(f\) and its inverse \(f^{-1}(y) = \frac{2}{\pi}\arctan(y)\) are continuous.
Thus, \(f\) is a homeomorphism, meaning \((-1, 1)\) and \(\mathbb{R}\) are topologically equivalent.
Notice a profound consequence: \(X\) is bounded, but \(Y\) is unbounded. Furthermore, \(Y\) is complete, but \(X\) is
not (e.g., the sequence \(x_n = 1 - 1/n\) is Cauchy on \(X\) but does not converge on \(X\)). This demonstrates that
topological equivalence preserves the "connectivity" and "openness" of a space, but does not necessarily
preserve metric properties like boundedness or completeness.
Why Uniform Equivalence Matters: Completeness Preservation
Example 2 above reveals a subtle but crucial point: a homeomorphism can destroy completeness.
The space \(\mathbb{R}\) is complete, yet it is topologically equivalent to the incomplete
\((-1, 1)\). This is exactly why uniform equivalence — which may look like a 'niche' middle
ground between topological and Lipschitz equivalence — plays a vital role in analysis.
A uniformly continuous map sends
Cauchy sequences
to Cauchy sequences, so a uniform equivalence preserves the
completeness
of a space. This makes uniform equivalence the standard sufficient condition
under which the convergence structure — and therefore the completion of a space — is preserved.
Topological equivalence alone is not enough.
Properties of a space that are preserved under homeomorphisms — such as
compactness and
connectedness — are called topological invariants.
These invariants will form the backbone of the upcoming manifold series: an \(n\)-dimensional topological manifold
is, by definition, a Hausdorff, second-countable topological space in which every point has a neighbourhood homeomorphic to an open subset of \(\mathbb{R}^n\).
Homeomorphism is therefore the very language in which manifolds are defined, and in turn the framework in which Geometric Deep Learning
formulates equivariance, invariance, and the geometric structure of data (graphs, 3D surfaces, Lie-group actions on feature spaces).