topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
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Kolmogorov space, Hausdorff space, regular space, normal space
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Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
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synthetic differential geometry
Introductions
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The magic algebraic facts
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Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Tietze extension theorem says that continuous functions extend from closed subsets of a normal topological space $X$ to the whole space $X$.
This is a close cousin of Urysohn's lemma with many applications.
One implication is that topological vector bundles over a topological space $X$ that trivialize over a closed subspace $A$ are equivalent to vector bundles on the quotient space $X/A$ (see there). This in turn is what implies the long exact sequence in cohomology for topological K-theory (see there).
For $X$ a normal topological space and $A \subset X$ a closed subspace, there is for every continuous function $f \colon A \to \mathbb{R}$ to the real line (with its Euclidean metric topology) a continuous function $\hat f \colon X \to \mathbb{R}$ extending it, i.e. such that $\hat f|_A = f$:
Therefore one also says that $\mathbb{R}$ is an absolute extensor in topology.
We produce a sequence of approximations to the desired extension by induction. Then we will observe that the sequence is a Cauchy sequence and conclude by observing that this implies that it limit is an extension of $f$ as desired.
For the induction step, let
be a continuous function on $X$ such that the difference of its restriction to $A$ with $f$ is a bounded function, for a bound $c_n \in (0,\infty) \subset \mathbb{R}$:
Consider then the pre-image subsets
Since the closed intervals $[-c_n,-c_n/3], [c_n/3, c_n] \subset \mathbb{R}$ are closed subsets, and since $f - \hat f_n\vert_A$ is a continuous function, these are closed subsets of $A$. Moreover, since subsets are closed in a closed subspace precisely if they are closed in the ambient space, these are also closed subsets of $X$.
Therefore, since $X$ is normal by assumption, it follows with Urysohn's lemma that there is a continuous function
with
and
Consider then the continuous function
This now satisfies
with
Moreover, observe that this function satisfies
To wit, this is because
for $a \in S_+$ we have $g_{n+1}(a) = \tfrac{c_n}{3}$ and $f(a) - \hat f_{n}(a) \in [c_n/3,c_n]$;
for $a \in S_-$ we have $g_{n+1}(a) = -\tfrac{c_n}{3}$ and $f(a) - \hat f_{n}(a) \in [-c_n/3,-c_n]$;
for $a \in Y \setminus \{S_+ \cup S_-\}$ we have $g(a) \in [-c_n/3,c_n/3]$ as well as $f(a) - \hat f_{n}(a) \in [-c_n/3, c_n/3]$.
It follows that if we set
then
This gives the induction step.
To start the induction, first assume that $f$ is bounded by a constant $c_0$. Then we may set
Hence induction now gives a sequence of continuous functions
with the property that
Moreover, for $n_1, n_2 \in \mathbb{N}$ with $n_2 \geq n_1$ and $x \in X$ we have
That the geometric series $\sum_{k = 0}^\infty 1/3^k$ converges
this becomes arbitrarily small for large $n_1$.
This means that the sequence $(\hat f_{n+1})_{n\in \mathbb{N}}$ is a Cauchy sequence in the supremum norm for real-valued functions.
Since uniform Cauchy sequences of continuous functions with values in a complete metric space converge uniformly to a continuous function (this prop.) this implies that the sequence converges uniformly to a continuous function. By construction, this is an extension as required.
Finally consider the case that $f$ is not a bounded function. In this case consider any homeomorphism $\phi \colon \mathbb{R}^1 \overset{\simeq}{\to} (-c_0,c_0) \subset \mathbb{R}^1$ between the real line and an open interval Then $\phi \circ f$ is a continous function bounded by $c_0$ and hence the above argument gives an extension $\widehat {\phi \circ f}$. Then $\phi^{-1} \circ \widehat{ \phi \circ f }$ is an extension of $f$.
See Whitney extension theorem, also Steenrod-Wockel approximation theorem.
Let $\mathbb{L} = (C^\infty Ring^{fin})^{op}$ be the category of smooth loci, the opposite category of finitely generated generalized smooth algebras. By the theorem discussed there, there is a full and faithful functor Diff $\hookrightarrow \mathbb{L}$.
For $A = C^\infty(\mathbb{R}^n)/J$ and $B = C^\infty(\mathbb{R}^n)/I$ with $I \subset J$ and $B \to A$ the projection of generalized smooth algebras the corresponding monomorphism $\ell A \to \ell B$ in $\mathbb{L}$ exhibits $\ell A$ as a closed smooth sublocus of $\ell B$.
Let $X$ be a smooth manifold and let $\{g_i \in C^\infty(X)\}_{i = 1}^n$ be smooth functions that are independent in the sense that at each common zero point $x\in X$, $\forall i : g_i(x)= 0$ we have the derivative $(d g_i) : T_x X \to \mathbb{R}^n$ is a surjection, then the ideal $(g_1, \cdots, g_n)$ coincides with the ideal of functions that vanish on the zero-set of the $g_i$.
This is lemma 2.1 in Chapter I of (MoerdijkReyes).
If $\ell A \hookrightarrow \ell B$ is a closed sublocus of $\ell B$ then every morphism $\ell A \to R$ extends to a morphism $\ell B \to R$
This is prop. 1.6 in Chapter II of (MoerdijkReyes).
Since we have $R = \ell C^\infty(\mathbb{R})$ and $C^\infty(\mathbb{R})$ is the free generalized smooth algebra on a single generator, a morphism $\ell A \to R$ is precisely an element of $C^\infty(\mathbb{R}^n)/J$. This is represented by an element in $C^\infty(\mathbb{R}^n)$ which in particular defines an element in $C^\infty(\mathbb{R}^n)/I$.
extension theorems | continuous functions | smooth functions |
---|---|---|
plain functions | Tietze extension theorem | Whitney extension theorem |
equivariant functions | equivariant Tietze extension theorem |
Leture notes include
Discussion via algebraic topology:
Discussion of the smooth version includes
See also
Last revised on November 17, 2020 at 16:15:58. See the history of this page for a list of all contributions to it.