Isometric embeddings of finite metric spaces

Oblakova, Anna

doi:10.3103/s0027132216010010

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Cantor Set and Totally Disconnected Fractals1

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2023

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“…In this regard, we noted that the totally disconnected fractal can be endowed with a suitable metric [229][230][231][232][233][234]. If this metric satisfies a stronger form of the triangle inequality (x, y) ≤ max{ (x, z), (z, y }∀x, y, z ∈ C, (52) then it is called an ultrametric or non-Archimedean metric [233].…”

Section: Cantor Set and Totally Disconnected Fractalsmentioning

confidence: 99%

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Ortíz

Martínez-Cruz

et al. 2023

Fractal Fract

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The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

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