# The dimension of X^n where X is a separable metric space

Fundamenta Mathematicae (1996)

• Volume: 150, Issue: 1, page 43-54
• ISSN: 0016-2736

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## Abstract

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For a separable metric space X, we consider possibilities for the sequence $S\left(X\right)={d}_{n}:n\in ℕ$ where ${d}_{n}=dim{X}^{n}$. In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is ${X}_{n}$ such that $S\left({X}_{n}\right)=n,n+1,n+2,...$, ${Y}_{n}$, for n >1, such that $S\left({Y}_{n}\right)=n,n+1,n+2,n+2,n+2,...$, and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of ${ℝ}^{2}$ is shown to exist which satisfies $1=dimX=dim{X}^{2}$ and $dim{X}^{3}=2$.

## How to cite

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Kulesza, John. "The dimension of X^n where X is a separable metric space." Fundamenta Mathematicae 150.1 (1996): 43-54. <http://eudml.org/doc/212162>.

@article{Kulesza1996,
abstract = {For a separable metric space X, we consider possibilities for the sequence $S(X) = \{d_n: n ∈ ℕ\}$ where $d_n = dim X^n$. In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is $X_n$ such that $S(X_n) = \{n, n+1, n+2,...\}$, $Y_n$, for n >1, such that $S(Y_n) = \{n, n+1, n+2, n+2, n+2,...\}$, and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of $ℝ^2$ is shown to exist which satisfies $1 = dim X = dim X^2$ and $dim X^3 = 2$.},
author = {Kulesza, John},
journal = {Fundamenta Mathematicae},
keywords = {separable metric space},
language = {eng},
number = {1},
pages = {43-54},
title = {The dimension of X^n where X is a separable metric space},
url = {http://eudml.org/doc/212162},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Kulesza, John
TI - The dimension of X^n where X is a separable metric space
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 1
SP - 43
EP - 54
AB - For a separable metric space X, we consider possibilities for the sequence $S(X) = {d_n: n ∈ ℕ}$ where $d_n = dim X^n$. In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is $X_n$ such that $S(X_n) = {n, n+1, n+2,...}$, $Y_n$, for n >1, such that $S(Y_n) = {n, n+1, n+2, n+2, n+2,...}$, and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of $ℝ^2$ is shown to exist which satisfies $1 = dim X = dim X^2$ and $dim X^3 = 2$.
LA - eng
KW - separable metric space
UR - http://eudml.org/doc/212162
ER -

## References

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1. [AK] R. D. Anderson and J. E. Keisler, An example in dimension theory, Proc. Amer. Math. Soc. 18 (1967), 709-713. Zbl0153.24602
2. [DRS] A. Dranishnikov, D. Repovš and E. Ščepin, On intersections of compacta of complementary dimensions in Euclidean space, Topology Appl. 38 (1991), 237-253. Zbl0719.54015
3. [E] R. Engelking, Dimension Theory, North-Holland, Amsterdam, 1978.
4. [H] Y. Hattori, Dimension and products of topological groups, Yokohama Math. J. 42 (1994), 31-40. Zbl0816.54025
5. [K] J. Krasinkiewicz, Imbeddings into ${ℝ}^{n}$ and dimension of products, Fund. Math. 133 (1989), 247-253. Zbl0715.54027
6. [Ku1] J. Kulesza, Dimension and infinite products in separable metric spaces, Proc. Amer. Math. Soc. 110 (1990), 557-563. Zbl0724.54032
7. [Ku2] J. Kulesza, A counterexample to the extension of a product theorem in dimension theory to the noncompact case, preprint.
8. [L] J. Luukkainen, Embeddings of n-dimensional locally compact metric spaces to 2n-manifolds, Math. Scand. 68 (1991), 193-209. Zbl0723.54013
9. [Sp] S. Spież, The structure of compacta satisfying dim(X × X) < 2 dim X, Fund. Math. 135 (1990), 127-145.

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