Alexandroff Plank

Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.

Definition

The construction of the Alexandroff plank starts by defining the topological space $(X,\tau)$ to be the Cartesian product of ,\omega_1/math> and 1,1 where $\omega_1$ is the first uncountable ordinal, and both carry the interval topology. The topology $\tau$ is extended to a topology $\sigma$ by adding the sets of the form
$U(\alpha,n) = \ \cup (\alpha,\omega_1] \times (0,1/n)$
where $p = (\omega_1,0) \in X.$

The Alexandroff plank is the topological space $(X,\sigma).$

It is called plank for being constructed from a subspace of the product of two spaces.

Properties

The space $(X,\sigma)$ has the following properties:
# It is Urysohn and completely Hausdorff spaces, Urysohn, since $(X,\tau)$ is regular. The space $(X,\sigma)$ is not regular, since $C = \$ is a closed set not containing $(\omega_1,0),$ while every neighbourhood of $C$ intersects every neighbourhood of $(\omega_1,0).$
# It is semiregular, since each basis rectangle in the topology $\tau$ is a regular open set and so are the sets $U(\alpha,n)$ defined above with which the topology was expanded.
# It is not countably compact, since the set $\$ has no upper limit point.
# It is not metacompact, since if $\$ is a covering of the ordinal space $$point-finite refinement, then the covering $\$ of $X$ defined by $U_1 = \ \cup ([0,\omega_1] \times (0,1]),$ $U_2 = [0,\omega_1] \times [-1,0),$ and $U_\alpha = V_\alpha \times [-1,1]$ has not point-finite refinement.

See also

*

References

* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
* S. Watson, ''The Construction of Topological Spaces''. Recent Progress in General Topology, Elsevier, 1992.

Topological spaces

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