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4. Topology of Metric Spaces 본문

General topology

4. Topology of Metric Spaces

woddlwoddl 2024. 5. 4. 14:17
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Topology of Metric Spaces

Topology of Metric Spaces

Metric Spaces and Their Topology

Metric Space: A metric space is a set \( X \) equipped with a distance function \( d: X \times X \rightarrow \mathbb{R} \) satisfying the following properties for all \( x, y, z \) in \( X \):

  1. Non-negativity: \d(x,y \geq 0 \) with equality if and only if \( x = y \).
  2. Identity of indiscernibles: \d(x,y = 0 \) if and only if \( x = y \).
  3. Symmetry: \d(x,y = dy,x \).
  4. Triangle inequality: \d(x,z \leq dx,y + dy,z \).
The topology induced by the metric \d\)consistsofopensetsdefinedbyopenballs\(B(x,r = \{ y \in X \mid dx,y < r \} \) for every \( x \) in \( X \) and \( r > 0 \).

Properties of Metric Spaces

Completeness: A metric space \( X \) is complete if every Cauchy sequence in \( X \) converges to a limit in \( X \).

Compactness: A metric space \( X \) is compact if every open cover of \( X \) has a finite subcover.

Connectedness: A metric space \( X \) is connected if it cannot be divided into two disjoint nonempty open sets.

Examples of Metric Spaces

Euclidean Space: The Euclidean space \Rn\)equippedwiththestandardEuclideanmetricisametricspace.Thedistancebetweentwopoints\(x=(x1,x2,,xn \) and \y=(y1,y2,,yn \) is given by \d(x,y = \sqrt{x1y1^2 + x2y2^2 + \ldots + xnyn^2} \).

Discrete Metric Space: Any set \X\)equippedwiththediscretemetric\(d(x,y = {0if x=y1if xy \) is a metric space.

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