Alexandrov topology

Builds on Topological space Open set Preorder Upper set

Definition

Let $X$ be a set and let $\tau$ be a topology on $X$ . The topology $\tau$ is an Alexandrov topology if for every family $(U_i)_{i \in I}$ of open sets of $\tau$ , the intersection $\bigcap_{i \in I} U_i$ is open. A topological space whose topology is an Alexandrov topology is called an Alexandrov space or an Alexandrov-discrete space.

Remarks

Equivalently, every point $x \in X$ has a smallest open set $U_x$ containing it, namely the intersection of all open sets that contain $x$ .

For a preorder $\le$ on $X$ , the family of all upper sets of $(X, \le)$ is an Alexandrov topology. Conversely, every Alexandrov topology arises in this way from exactly one preorder, its specialization preorder.

Example: every topology on a finite set is an Alexandrov topology, because then every family of open sets is finite.

Non-example: the usual topology of $\R$ is not an Alexandrov topology. The open sets $(-1/n, 1/n)$ for $n \ge 1$ have intersection $\set{0}$ , which is not open.

The name "Alexandrov space" is also used in metric geometry for spaces with curvature bounded below; that notion is unrelated.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Open set concept
Preorder concept
Topological space concept
Upper set concept
Alexandrov topology definition

Tag	`CXZZJ1QV5K`
URI	`m2://CXZZJ1QV5K`
Kind	definition
Writing	W0 stub
MSC	54A05, 06A06
Defines	Alexandrov topology
Uses	Topological space, Open set, Preorder, Upper set

Alexandrov topology

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here