nLab
perfect space
Context
Topology
topology (point-set topology , point-free topology )

see also differential topology , algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

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

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

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

Analysis Theorems

topological homotopy theory

Contents
Idea
Georg Cantor used to define the continuum as a perfect space that is connected as well. Hence, the property of ‘being perfect’, as the name indicates as well, can be viewed as forming part of the concept of a ‘prototypical’ topological space .

Definition
A topological space $X$ is perfect if it has no isolated points, i.e., if every point $x$ belongs to the topological closure of its complement $X \setminus \{x\}$ .

In a topological space $X$ , a subset is said to be perfect if it is closed in $X$ and perfect in its subspace topology .

Example
Properties
Every topological space $X$ is the disjoint union of a scattered subset and a perfect subset.
Reference
S. Willard, General Topology , Addison-Wesley Reading 1970. (Dover reprint 2004, pp.216ff)
Last revised on May 9, 2017 at 02:59:12.
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