G-delta subspace

**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
* convenient category of topological spaces
**[Universal constructions](Top#UniversalConstructions)**
* initial topology, final topology
* subspace, quotient space,
* fiber space, space attachment
* product space, disjoint union space
* mapping cylinder, mapping cocylinder
* mapping cone, mapping cocone
* mapping telescope
* colimits of normal spaces
**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
* real line, plane
* 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
* loop space, path space
* 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**
* Hausdorff spaces are sober
* schemes are sober
* continuous images of compact spaces are compact
* closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
* open subspaces of compact Hausdorff spaces are locally compact
* quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff
* compact spaces equivalently have converging subnet of every net
* Lebesgue number lemma
* sequentially compact metric spaces are equivalently compact metric spaces
* compact spaces equivalently have converging subnet of every net
* sequentially compact metric spaces are totally bounded
* continuous metric space valued function on compact metric space is uniformly continuous
* paracompact Hausdorff spaces are normal
* paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
* closed injections are embeddings
* proper maps to locally compact spaces are closed
* injective proper maps to locally compact spaces are equivalently the closed embeddings
* locally compact and sigma-compact spaces are paracompact
* locally compact and second-countable spaces are sigma-compact
* second-countable regular spaces are paracompact
* CW-complexes are paracompact Hausdorff spaces
**Theorems**
* Urysohn's lemma
* Tietze extension theorem
* Tychonoff theorem
* tube lemma
* Michael's theorem
* Brouwer's fixed point theorem
* topological invariance of dimension
* Jordan curve theorem
**Analysis Theorems**
* Heine-Borel theorem
* intermediate value theorem
* extreme value theorem
**topological homotopy theory**
* left homotopy, right homotopy
* homotopy equivalence, deformation retract
* fundamental group, covering space
* fundamental theorem of covering spaces
* homotopy group
* weak homotopy equivalence
* Whitehead's theorem
* Freudenthal suspension theorem
* nerve theorem
* homotopy extension property, Hurewicz cofibration
* cofiber sequence
* Strøm model category
* classical model structure on topological spaces

A **G-delta**, $G_\delta$, subset of a topological space is a set that can be written as the intersection of a countable family of open sets.

One place where $G_\delta$-subsets occur is when looking at continuous maps from an arbitrary topological space to a metric space (or, more generally, a first countable space). In particular, when considering continuous real-valued functions. Thus we have the following connections to the separation axioms.

A normal space in which every closed set is a $G_\delta$-set is perfectly normal.

In a completely regular space, every singleton set that is a $G_\delta$-set is the unique global maximum of a continuous real-valued function.

One direction is obvious. For the other, let $v$ be a point in a completely regular space $X$ such that $\{v\}$ is a $G_\delta$-set. Let $\{V_n\}$ be a sequence of open sets such that $\bigcap V_n = \{v\}$. We now define a sequence of functions $(f_n)$ recursively with the properties:

- $f_n \colon X \to [0,1]$ is a continuous function,
- $f_n(v) = 1$,
- $f_n^{-1}(1)$ is a neighbourhood of $v$,
- for $n \gt 1$, $f_n$ has support in $V_n \cap f_{n-1}^{-1}(1)$ whilst $f_1$ has support in $V_1$,

Having defined $f_1, \dots, f_{n-1}$, we define $f_n$ as follows. Since $V_n \cap f_{n-1}^{-1}(1)$ is a neighbourhood of $v$ and $X$ is completely regular, there is a continuous function $\tilde{f}_n \colon X \to [0,1]$ with support in this neighbourhood and such that $\tilde{f}_n(v) = 1$. We then compose with a continuous, increasing surjection $[0,1] \to [0,1]$ which maps $[\frac12,1]$ to $1$. The resulting function is the required $f_n$.

We then define a function $f \colon X \to [0,1]$ by

$f(x) \coloneqq \sum_{n=1}^\infty \frac1{2^n} f_n(x).$

By construction, $f^{-1}(1) = \{v\}$.

We need to prove that this is continuous. First, note that if $f_n(x) \ne 0$ then $f_k(x) = 1$ for $k \lt n$ and if $f_n(x) \ne 1$ then $f_k(x) = 0$ for $k \gt n$. Hence the preimage under $f$ of $(\frac{2^k-1}{2^k}, \frac{2^{k+1}-1}{2^{k+1}})$ is $f_n^{-1}(0,1)$ and $f$ restricted to this preimage is a scaled translate of $f_n$. From this, we deduce that the preimage of any open set not containing $1$ is open. Thus $f$ is continuous everywhere except possibly at $v$. Continuity at $v$ is similarly simple: given a set of the form $(1 -\epsilon,1]$ then there is some $n$ such that $2^{-n} \lt \epsilon$, whence $f^{-1}(1-\epsilon,1]$ contains all points such that $f_k(x) = 1$ for $k \le n$, which by construction is a neighbourhood of $v$. Hence $f$ is continuous and has a single global maximum at $v$.

Revised on June 22, 2010 23:27:04
by Toby Bartels
(75.88.99.206)