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relatively prime integer topology
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(Definition)
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Let $X$ be the set of strictly positive integers. The relatively prime integer topology on $X$ is the topology determined by a basis consisting of the sets
for any $a$ and $b$ are relatively prime integers. That this does indeed form a basis is found in this entry.
Equipped with this topology, $X$ is $T_0$ , $T_1$ ,and $T_2$ , but satisfies none of the higher separation axioms (and hence meet very few compactness criteria).
We can define a coarser topology on $X$ by considering the subbasis of the above basis consisting of all $U(a,b)$ with $a$ being a prime. This is called the prime integer topology on $\Z^+$ .
- 1
- L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
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"relatively prime integer topology" is owned by mathcam.
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| Also defines: |
prime integer topology |
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Cross-references: prime, subbasis, coarser, compactness, meet, separation axioms, relatively prime, basis, topology, integers, positive, strictly
This is version 4 of relatively prime integer topology, born on 2004-10-07, modified 2007-01-04.
Object id is 6314, canonical name is RelativelyPrimeIntegerTopology.
Accessed 3164 times total.
Classification:
| AMS MSC: | 54E30 (General topology :: Spaces with richer structures :: Moore spaces) |
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Pending Errata and Addenda
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