nLab
frame of opens
Context
Topology
topology (point-set topology )

see also algebraic topology , functional analysis and homotopy theory

Introduction

Basic concepts
open subset , closed subset , neighbourhood

topological space (see also locale )

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

topological vector space , Banach space , Hilbert space

topological group

topological manifold

cell complex , CW-complex

Examples
empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

sphere , ball ,

circle , torus , annulus

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements
Theorems
Theorems

Topos Theory
topos theory

Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
Given a topological space $X$ , the open subspaces of $X$ form a poset which is in fact a frame . This is the frame of open subspaces of $X$ . When thought of as a locale , this is the topological locale $\Omega(X)$ . When thought of as a category , this is the category of open subsets of $X$ .

Similarly, given a locale $X$ , the open subspaces of $X$ form a poset which is in fact a frame. This is the frame of open subspaces of $X$ . When thought of as a locale, this is simply $X$ all over again. When thought of as a category, this is a site whose topos of sheaves is a localic topos .

The frame of open subsets of the point is given by the power set of a singleton, or more generally by the object of truth values of the ambient topos.