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4. Topology of Metric Spaces 본문
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 \( X \):
- Non-negativity: \d(x,y \geq 0 \) with equality if and only if \( x = y \).
- Identity of indiscernibles: \d(x,y = 0 \) if and only if \( x = y \).
- Symmetry: \d(x,y = dy,x \).
- Triangle inequality: \d(x,z \leq dx,y + dy,z \).
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{x1−y1^2 + x2−y2^2 + \ldots + xn−yn^2} \).
Discrete Metric Space: Any set \X\)equippedwiththediscretemetric\(d(x,y = {0if x=y1if x≠y \) is a metric space.
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