Metrics, metric spaces, and the metric topology

Overview §

Definition: Metric, metric space

A metric on a set $X$ is a function $d:X\times X\to \mathbb{R}$ that satisfies the following conditions for all $x,y,z\in X$:

• (M1) Non-negativity and identity. $d(x,y)\ge 0$, with $d(x,y)=0$ if and only if $x=y$;
• (M2) Symmetry. $d(x,y)=d(y,x)$;
• (M3) Triangle inequality. We have $d(x,y)\le d(x,z)+d(z,y)$.

The value $d(x,y)$ is often called the distance between $x$ and $y$ in the metric $d$. A metric space is a pair $(X,d)$ where $X$ is any set and $d$ is a metric on $X$.

Some properties of a metric space, such as boundedness, are not entirely topological, but depend on a choice of metric.

Related: Inner product spaces

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Distances §

Definition: Distance between a set and a point in a metric space

Let $(X,d)$ be a metric space. If $A\subseteq X$ is nonempty and $x\in X$, the distance from $x$ to $A$ is

$$d(x,A)=\inf \{d(x,a)|a\in A\}.$$

Further, the map from $X\to \mathbb{R}$ defined by $x\mapsto d(x,A)$ for all $x\in X$ is continuous.

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Elements and subsets of metric spaces §

Definition (Rudin 2.18): Elements and subsets of metric spaces

Let $(X,d)$ be a metric space.

• A neighborhood of a point $p\in X$ is the set

$$B_{ϵ}(p)=\{q\in X|d(p,q)<r\}$$

for some radius $r>0$.

• A point $p$ is a limit point of the set $E\subseteq X$ if every neighborhood of $p$ contains a point $q\in E$ where $q\ne p$; that is, every neighborhood of $p$ intersects $E$ at some point other than itself.
• A point $p$ is an interior point of $E$ if there exists a neighborhood $U$ of $p$ such that $U\subseteq E$.
• $E$ is closed if every limit point of $E$ is a point of $E$, and open if every point of $E$ is an interior point of $E$.
• $E$ is dense in $X$ if every point of $X$ is either a limit point of $E$, in $E$, or both.

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The metric topology §

Metrics generate topological bases, which then generate a topology.

Definition: Metric topology, metrizable space

If $(X,d)$ is a metric space (i.e., $d$ is a metric on the set $X$), for each $x\in X$, we define the $ϵ$-ball about $x$ as the set

$$B_d(x,ϵ)=\{y\in X|d(x,y)<ϵ.\}$$

Then the set of all open balls

$$\{B_d(x,ϵ)|x\in X,ϵ>0\}$$

is a basis for the metric topology induced by $d$. A general topological space $(X,\mathcal{T})$ is metrizable if there exists some metric on $X$ which induces $\mathcal{T}$.

The notation $B_d$ emphasizes that the set of open balls is inducing the metric topology with respect to the specific metric $d$.

Lemma: Every metrizable space is Hausdorff

If $X$ is a metrizable space and has a topology induced by the metric $d$, then $X$ is Hausdorff.

Lemma: Using bases to determine which metric topology is finer

Let $d,d’$ be metrics on $X$ with induced topologies $\mathcal{T},\mathcal{T}’$ respectively. Then $T’$ is finer than $\mathcal{T}$ (i.e., $\mathcal{T}\subseteq T’$) if and only if for all $x\in X$ and all $ϵ>0$, there exists $\delta >0$ such that $B_{d’}(x,\delta )\subseteq B_d(x,ϵ)$.

The typical $ϵ-\delta$ and convergent sequence definitions of continuity also hold for general metrizable spaces.

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Examples §

Metric generating the discrete topology §

Transclude of Discrete-topology#^6abe9b

Euclidean metric §

The Euclidean metric on ${\mathbb{R}}^n$ is the norm $|x-y|$ defined by

$$d(x,y)=\sqrt{(x_1-y_1{)}^2+\cdots +(x_n-y_n{)}^2}.$$

“Square metric” on ${\mathbb{R}}^n$ §

The “square metric” on ${\mathbb{R}}^n$ is defined by

$$\rho (x,y)=\text{max}\{|x_1-y_1|,...,|x_n-y_n|\}.$$

Theorem: The Euclidean and the square metrics induce the same topology

The Euclidean and the square metrics induce the same topology on ${\mathbb{R}}^n$ (and both are equal to the product topology).

Standard bounded metric §

Definition: Standard bounded metric

Given any metric space $(X,d)$, the standard bounded metric corresponding to $d$ is defined as

$$\overline{d}(x,y)=\text{min}\{d(x,y),1\}.$$

Theorem: The standard bounded metric induces the same topology as the original

For any metric space $(X,d)$:

1. The standard bounded metric $\overline{d}$ induces the same topology as $d$;
2. Every subset $A\subseteq X$ is bounded with respect to $\overline{d}$.

Intuitively, we can show (1) by using balls of radius $<1$ to generate the topology induced by $d$. Statement (2) gives an example of how boundedness is a property of the metric, and not a general topological property; the result is immediate by definition of $\overline{d}$.

#wip Upshot: boundedness is a property of….?

Uniform metric on the general product §

Definition: Uniform metric, uniform topology

Consider the product ${\prod }_J\mathbb{R}$, where $J$ is any indexing set. As a set, ${\prod }_J\mathbb{R}$ is the set of all functions $J\to \mathbb{R}$, so we may also denote it ${\mathbb{R}}^J$. The uniform metric on ${\mathbb{R}}^J$ is defined by

$$\overline{\rho }(x,y)=\text{sup}\{\overline{d}(x_{\alpha },y_{\alpha })|\alpha \in J\},$$

where $\overline{d}(x,y)=\text{min}\{d(x,y),1\}$ is the standard bounded metric on $\mathbb{R}$ and $d(x,y)=|x-y|$ is the standard metric. It induces the uniform topology on ${\mathbb{R}}^J$.

• use $\overline{d}$ rather than $d$ to avoid infinity (e.g., what happens if $J={\mathbb{Z}}_{>0}$)

supremum

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Review §

Topology

• Show that the basis for the metric topology is indeed a basis. ⭐
• Check the example metrics are indeed metrics, particularly $\overline{d}$. ⭐
• Show that every metrizable space is Hausdorff. (Hint: select radius $ϵ>0$ to create disjoint open balls in $X$.)
• Prove that the Euclidean and the square metrics induce the same topology on ${\mathbb{R}}^n$. (Hint: apply the lemma about using bases to determine which metric topology is finer.) ⭐
• Prove that the standard bounded metric induces the same topology as the original metric. ⭐
• Show that the uniform metric on the general product $\overline{\rho }$ is a metric. ⭐

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Proof appendix §

Theorem: The Euclidean and the square metrics induce the same topology

The Euclidean and the square metrics induce the same topology on ${\mathbb{R}}^n$ (and both are equal to the product topology).

Link to original

Sketch from Topology.

1. Verify the following inequality: $\rho (x,y)\le d(x,y)\le \sqrt{n}\cdot \rho (x,y)$.
2. $\rho (x,y)\le d(x,y)⟹$ $y\in B_d(x,ϵ)\subseteq B_{\rho }(x,ϵ)$ for all $x\in X$, and the inequality is bounded above by $ϵ>0$. #concept-question for what epsilon?

Definition: Distance between a set and a point in a metric space

Let $(X,d)$ be a metric space. If $A\subseteq X$ is nonempty and $x\in X$, the distance from $x$ to $A$ is

$$d(x,A)=\inf \{d(x,a)|a\in A\}.$$

Further, the map from $X\to \mathbb{R}$ defined by $x\mapsto d(x,A)$ for all $x\in X$ is continuous.

Link to original

#wip