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
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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
Analysis Theorems
($\Delta$-Generated spaces)
A $\Delta$-generated space (alias numerically generated space) (Smith, Dugger 03) is a topological space $X$ whose topology is the final topology induced by all continuous functions of the form $\Delta^n_{top} \to X$, hence whose domain $\Delta^n_{top}$ is one of the standard topological simplices, for $n \in \mathbb{N}$.
A morphism between $\Delta$-generated spaces is just a continuous function, hence the category of $\Delta$-generated spaces is the full subcategory on these spaces inside all TopologicalSpaces,
(as colimits of topological simplices)
Equivalently, the class of $\Delta$-generated spaces is the closure of the set of topological simplices $\Delta^n_{top}$ under small colimits in topological spaces (see at Top – universal constructions).
(as Euclidean-generated spaces)
For each $n$ the topological simplex $\Delta^n$ is a retract of the ambient Euclidean space/Cartesian space $\mathbb{R}^n$ (as a non-empty convex subset of a Euclidean space it is in fact an absolute retract). Hence the identity function on $\Delta^n$ factors as
It follows that every continuous function $f$ with domain the topological simplex extends as a continuous function to Euclidean space:
Therefore the condition that a topological space $X$ be $\Delta$-generated (Def. ) is equivalent to saying that its topology is final with respect to all continuous functions $\mathbb{R}^n \to X$ out of Euclidean/Cartesian spaces.
(as D-topological spaces)
By Prop. below, the Euclidean-generated spaces and hence, by Remark , the $\Delta$-generated spaces, are equivalently those that arise from equipping a diffeological space with its D-topology, hence are, in this precise sense, the D-topological spaces. Luckily, “D-topological space” may also serve as an abbreviation for “Delta-generated topological space”.
(Euclidean-generated spaces are convenient)
The category of Euclidean-generated spaces/$\Delta$-generated spaces (Def. ) is a convenient category of topological spaces in that:
it contains all CW-complexes (SYH 10, Cor. 4.4),
it is complete and cocomplete category (SYH 10, Prop. 3.4),
it is locally presentable (FR 08, Cor. 3.7)
it is cartesian closed (SYH 10, Cor. 4.6):
its mapping space internal homs $Maps(X,Y)$ are given by the D-topology of the internal homs in diffeological spaces (i.e. in presheaves) between continuous diffeologies (SYH 10, Prop. 4.7):
(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 was a gap in the original proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$. The gap is claimed to be filled now, 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.
(diffeological singular simplicial set)
Consider the simplicial diffeological space
which in degree $n$ is the standard extended n-simplex inside Cartesian space $\mathbb{R}^{n+1}$, equipped with its sub-diffeology.
This induces a nerve and realization adjunction between diffeological spaces and simplicial sets:
where the right adjoint is the diffeological singular simplicial set functor $Sing_{diff}$.
(e.g. Christensen-Wu 13, Def. 4.3)
(diffeological singular simplicial set as path ∞-groupoid)
Regarding simplicial sets as presenting ∞-groupoids, we may think of $Sing_{diff}(X)$ (Def. ) as the path ∞-groupoid of the diffeological space $X$.
In fact, by the discussion at shape via cohesive path ∞-groupoid we have that $Sing_{diff}$ is equvialent to the shape of diffeological spaces regarded as objects of the cohesive (∞,1)-topos of smooth ∞-groupoids:
(topological homotopy type is cohesive shape of continuous diffeology)
For every $X \in$ TopologicalSpaces, the cohesive shape/path ∞-groupoid presented by its diffeological singular simplicial set (Def. , Remark ) of its continuous diffeology is equivalent to the homotopy type of $X$ presented by the ordinary singular simplicial set:
The category of $\Delta$-generated spaces carries the structure of a cofibrantly generated model category with the same generating (acyclic) cofibrations as for the classical model structure on topological spaces and such that the coreflection into all TopologicalSpaces (Prop. ) is a Quillen equivalence to the classical model structure on topological spaces.
$\Delta$-generated spaces were originally proposed by Jeff Smith as a nice category of spaces for homotopy theory.
Jeff Smith, A really convenient category of topological spaces (unpublished and possibly non-existent, according to Dugger 03, p. 4)
Daniel Dugger, Notes on Delta-generated spaces, 2003 (web, pdf)
A proof that the category of $\Delta$-generated spaces is locally presentable is in:
See also at directed homotopy theory.
The model structure on Delta-generated topological spaces Quillen equivalent to the classical model structure on topological spaces is due to
Relation to diffeological spaces:
Kazuhisa Shimakawa, K. Yoshida, Tadayuki Haraguchi, Homology and cohomology via enriched bifunctors, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 (arXiv:1010.3336, doi:10.2206/kyushujm.72.239)
J. Daniel Christensen, Gord Sinnamon, Enxin Wu, The $D$-topology for diffeological spaces, Pacific Journal of Mathematics 272 (1), 87-110, 2014 (arXiv:1302.2935, doi:10.2140/pjm.2014.272.87)
Last revised on April 15, 2021 at 14:17:27. See the history of this page for a list of all contributions to it.