Continuous images of Cantor's ternary set

Fabian Dreher; Tony Samuel

doi:10.4169/amer.math.monthly.121.07.640

Continuous images of Cantor's ternary set

Fabian Dreher, Tony Samuel

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.

Original language	English
Pages (from-to)	640-643
Number of pages	4
Journal	The American Mathematical Monthly
Volume	121
Issue number	7
DOIs	https://doi.org/10.4169/amer.math.monthly.121.07.640
Publication status	Published - Aug 2014

Bibliographical note

Translated into Mandarin: Shu Xue Yi Lin 35(4), 381-384 (2016).

Keywords

Cantor sets
Continuity
Topological spaces

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https://doi.org/10.4169/amer.math.monthly.121.07.640Licence: None: All rights reserved

Cite this

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title = "Continuous images of Cantor's ternary set",

abstract = "The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.",

keywords = "Cantor sets, Continuity, Topological spaces",

author = "Fabian Dreher and Tony Samuel",

note = "Translated into Mandarin: Shu Xue Yi Lin 35(4), 381-384 (2016).",

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AU - Samuel, Tony

N1 - Translated into Mandarin: Shu Xue Yi Lin 35(4), 381-384 (2016).

PY - 2014/8

Y1 - 2014/8

N2 - The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.

AB - The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.

KW - Cantor sets

KW - Continuity

KW - Topological spaces

UR - https://arxiv.org/abs/1303.3810

U2 - 10.4169/amer.math.monthly.121.07.640

DO - 10.4169/amer.math.monthly.121.07.640

M3 - Article

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VL - 121

SP - 640

EP - 643

JO - The American Mathematical Monthly

JF - The American Mathematical Monthly

IS - 7

ER -

Continuous images of Cantor's ternary set

Abstract

Bibliographical note

Keywords

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Cite this