Axiom of limitation of size

In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names:

• The odd–even topology is the topology where $X=\mathbb{\mathbb{N}}$ and $P=\{\{2k-1,2k\},k\in \mathbb{\mathbb{N}}\}.$
• The deleted integer topology is defined by letting $X=\begin{array}{c}\bigcup _{n\in \mathbb{\mathbb{N}}}(n-1,n)\subset \mathbb{ℝ}\end{array}$ and $P=\{(0,1),(1,2),(2,3),\dots \}$.

The trivial partitions yield the discrete topology (each point of X is a set in P) or indiscrete topology ($P=\{X\}$).

Any set X with a partition topology generated by a partition P can be viewed as a pseudometric space with a pseudometric given by:

$d(x,y)=\{\begin{array}{ll}0& \text{if }x\text{ and }y\text{ are in the same partition}\\ 1& \text{otherwise},\end{array}$

This is not a metric unless P yields the discrete topology.

The partition topology provides an important example of the independence of various separation axioms. Unless P is trivial, at least one set in P contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X is not a Kolmogorov space, nor a T₁ space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, X is a regular, completely regular, normal and completely normal.

We note also that X/P is the discrete topology.

References

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