Abstract
We shall shortly show that a non-empty compact Hausdorff space X is zero-dimensional iff X is totally disconnected. Recall that \( \mathop {\mathrm {comp}} \nolimits (x)\), the (connected) component of a point x ∈ X, is the union of all connected subspaces of X that contain x. The intersection of all clopen sets of X that contain x, denoted here by \( \mathop {\mathrm {qcomp}} \nolimits (x)\), is called the quasi-component of x. If \( \mathop {\mathrm {qcomp}} \nolimits (x) = \{x\}\) for every x ∈ X, X is called totally disconnected. If \( \mathop {\mathrm {comp}} \nolimits (x) = \{x\}\) for every x ∈ X, X is called hereditarily disconnected. Note that both \( \mathop {\mathrm {comp}} \nolimits (x)\) and \( \mathop {\mathrm {qcomp}} \nolimits (x)\) are closed subsets of X and \( \mathop {\mathrm {comp}} \nolimits (x)\) is connected.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P. Erdős, The dimension of the rational points in Hilbert space. Ann. Math. 41, 734–736 (1940)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Charalambous, M.G. (2019). Connected Components and Dimension. In: Dimension Theory. Atlantis Studies in Mathematics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-22232-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-22232-1_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22231-4
Online ISBN: 978-3-030-22232-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)