nLab
G-delta subspace

**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 * 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 ## 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 * 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 * real line, plane * sphere, ball, * circle, torus, annulus * polytope, polyhedron * projective space (real, complex) * classifying space * configuration space * 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 * CW-complexes are paracompact Hausdorff spaces * subsets are closed in a closed subspace precisely if they are closed in the ambient space * paracompact Hausdorff spaces are normal * 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 * 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 * second-countable regular spaces are paracompact ## Theorems * Urysohn's lemma * Tietze extension theorem * Tychonoff theorem * tube lemma * Heine-Borel theorem * Michael's theorem * Brouwer's fixed point theorem * topological invariance of dimension * Jordan curve theorem ## Basic **homotopy theory** * homotopy group * covering space **Theorems** * Whitehead's theorem * Freudenthal suspension theorem * nerve theorem

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Contents

Definition

Definition

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

Results

One place where G δG_\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.

Theorem

A normal space in which every closed set is a G δG_\delta-set is perfectly normal.

Theorem

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

Proof

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

  1. f n:X[0,1]f_n \colon X \to [0,1] is a continuous function,
  2. f n(v)=1f_n(v) = 1,
  3. f n 1(1)f_n^{-1}(1) is a neighbourhood of vv,
  4. for n>1n \gt 1, f nf_n has support in V nf n1 1(1)V_n \cap f_{n-1}^{-1}(1) whilst f 1f_1 has support in V 1V_1,

Having defined f 1,,f n1f_1, \dots, f_{n-1}, we define f nf_n as follows. Since V nf n1 1(1)V_n \cap f_{n-1}^{-1}(1) is a neighbourhood of vv and XX is completely regular, there is a continuous function f˜ n:X[0,1]\tilde{f}_n \colon X \to [0,1] with support in this neighbourhood and such that f˜ n(v)=1\tilde{f}_n(v) = 1. We then compose with a continuous, increasing surjection [0,1][0,1][0,1] \to [0,1] which maps [12,1][\frac12,1] to 11. The resulting function is the required f nf_n.

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

f(x) n=1 12 nf n(x). f(x) \coloneqq \sum_{n=1}^\infty \frac1{2^n} f_n(x).

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

We need to prove that this is continuous. First, note that if f n(x)0f_n(x) \ne 0 then f k(x)=1f_k(x) = 1 for k<nk \lt n and if f n(x)1f_n(x) \ne 1 then f k(x)=0f_k(x) = 0 for k>nk \gt n. Hence the preimage under ff of (2 k12 k,2 k+112 k+1)(\frac{2^k-1}{2^k}, \frac{2^{k+1}-1}{2^{k+1}}) is f n 1(0,1)f_n^{-1}(0,1) and ff restricted to this preimage is a scaled translate of f nf_n. From this, we deduce that the preimage of any open set not containing 11 is open. Thus ff is continuous everywhere except possibly at vv. Continuity at vv is similarly simple: given a set of the form (1ϵ,1](1 -\epsilon,1] then there is some nn such that 2 n<ϵ2^{-n} \lt \epsilon, whence f 1(1ϵ,1]f^{-1}(1-\epsilon,1] contains all points such that f k(x)=1f_k(x) = 1 for knk \le n, which by construction is a neighbourhood of vv. Hence ff is continuous and has a single global maximum at vv.

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