Kolmogorov topological space in nLab

Contents

Definition

A topological space $(X,\tau)$ is called a Kolmogorov space if it satisfies the $T_0$-separation axiom, hence if for $x_1 \neq x_2 \in X$ any two distinct points, then at least one of them has an open neighbourhood $U_{x_i} \in \tau$ which does not contain the other point. This is equivalent to its contrapositive: for all points $x_1, x_2 \in X$, if every open neighbourhood $U_{x_i} \in \tau$ which contains one of the points also contains the other point, then the two points are equal $x_1 = x_2$.

Properties

Specialization order

Every topological space $(X, O(X))$ is a preorder with respect to the specialization order $\forall_{U:O(X)} (x \in U) \implies (y \in U)$.

The proof follows from the contrapositive of the definition of a $T_0$-space.

Alternative Characterizations

Regard Sierpinski space $\Sigma$ as a frame object in the category of topological spaces (meaning: the representable functor $Top(-, \Sigma): Top^{op} \to Set$ lifts through the monadic forgetful functor $Frame \to Set$), hence as a dualizing object that induces a contravariant adjunction between frames and topological spaces. Thus for a space $X$, the frame $Top(X, \Sigma)$ is the frame of open sets; for a frame $A$, there is an accompanying topological space of points $Frame(A, \Sigma)$, a subspace of the product space $\Sigma^{{|A|}}$. The unit of the adjunction is called the double dual embedding.

The proof is trivial: the monomorphism condition translates to saying that for any points $x, y \in X$, if the truth values of $x \in U$ and $y \in U$ agree for every open set $U$, then $x = y$.

Reflection

• Hausdorff topological space

• nice topological space

preorder	partial order	equivalence relation	equality
topological space	Kolmogorov topological space	symmetric topological space	accessible topological space

References

• Wikipedia, Kolmogorov space