topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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fiber space, space attachment
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Kolmogorov space, Hausdorff space, regular space, normal space
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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
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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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?,
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Given a diffeological space, the D-topology (“diffeological topology”) is a natural topology on its underlying set of points, namely the final topology which makes all its plots be continuous functions.
(adjunction between topological spaces and diffeological spaces)
There is a pair of adjoint functors
between the categories of TopologicalSpaces and of DiffeologicalSpaces, where
$Cdfflg$ takes a topological space $X$ to the continuous diffeology, namely the diffeological space on the same underlying set $X_s$ whose plots $U_s \to X_s$ are the continuous functions (from the underlying topological space of the domain $U$).
$Dtplg$ takes a diffeological space to the diffeological topology (D-topology), namely the topological space with the same underlying set $X_s$ and with the final topology that makes all its plots $U_{s} \to X_{s}$ into continuous functions: called the D-topology.
Hence a subset $O \subset \flat X$ is an open subset in the D-topology precisely if for each plot $f \colon U \to X$ the preimage $f^{-1}(O) \subset U$ is an open subset in the Cartesian space $U$.
Moreover:
the fixed points of this adjunction $X \in$TopologicalSpaces (those for which the counit is an isomorphism, hence here: a homeomorphism) are precisely the Delta-generated topological spaces (i.e. D-topological spaces):
this is an idempotent adjunction, which exhibits $\Delta$-generated/D-topological spaces as a reflective subcategory inside diffeological spaces and a coreflective subcategory inside all topological spaces:
Finally, these adjunctions are a sequence of Quillen equivalences with respect to the:
classical model structure on topological spaces | model structure on D-topological spaces | model structure on diffeological spaces |
Caution: There may be a gap in the proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$, see the commented references here.
These adjunctions and their properties are observed in Shimakawa-Yoshida-Haraguchi 10, Prop. 3.1, Prop. 3.2, Lemma 3.3. The model structures and Quillen equivalences are due to Haraguchi 13, Thm. 3.3 (on the left) and Haraguchi-Shimakawa 13, Sec. 7 (on the right, but this may have a gap).
We spell out the existence of the idempotent adjunction (1):
First, to see we have an adjunction $Dtplg \dashv Cdfflg$, we check the hom-isomorphism (here).
Let $X \in DiffeologicalSpaces$ and $Y \in TopologicalSpaces$. Write $(-)_s$ for the underlying sets. Then a morphism, hence a continuous function of the form
is a function $f_s \colon X_s \to Y_s$ of the underlying sets such that for every open subset $A \subset Y_s$ and every smooth function of the form $\phi \colon \mathbb{R}^n \to X$ the preimage $(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n$ is open. But this means equivalently that for every such $\phi$, $f \circ \phi$ is continuous. This, in turn, means equivalently that the same underlying function $f_s$ constitutes a smooth function $\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y)$.
In summary, we thus have a bijection of hom-sets
given simply as the identity on the underlying functions of underlying sets. This makes it immediate that this hom-isomorphism is natural in $X$ and $Y$ and this establishes the adjunction.
Next, to see that the D-topological spaces are the fixed points of this adjunction, we apply the above natural bijection on hom-sets to the case
to find that the counit of the adjunction
is given by the identity function on the underlying sets $(\epsilon_X)_s = id_{(X_s)}$.
Therefore $\eta_X$ is an isomorphism, namely a homeomorphism, precisely if the open subsets of $X_s$ with respect to the topology on $X$ are precisely those with respect to the topology on $Dtplg(Cdfflg(X))$, which means equivalently that the open subsets of $X$ coincide with those whose pre-images under all continuous functions $\phi \colon \mathbb{R}^n \to X$ are open. This means equivalently that $X$ is a D-topological space.
Finally, to see that we have an idempotent adjunction, we check that the comonad
is an idempotent comonad, hence that
is a natural isomorphism. But, as before for the adjunction counit $\epsilon$, we have that also the adjunction unit $\eta$ is the identity function on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.
The concept originates with
The relation to Delta-generated topological spaces is due to
Discussion of the D-topology for mapping spaces:
Last revised on June 7, 2020 at 08:06:26. See the history of this page for a list of all contributions to it.