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
In the following we say Top-enriched category and Top-enriched functor etc. for what often is referred to as “topological category” and “topological functor” etc. As discussed there, these latter terms are ambiguous.
Write
for the full subcategory of Top on the compactly generated topological spaces. Under forming Cartesian product
and compactly generated mapping spaces
this is a cartesian closed category (see at convenient category of topological spaces).
A topologically enriched category $\mathcal{C}$ is a $Top_{cg}$-enriched category, hence:
for each $a,b\in Obj(\mathcal{C})$ a compactly generated topological space
called the space of morphisms or the hom-space between $a$ and $b$;
for each $a,b,c\in Obj(\mathcal{C})$ a continuous function
out of the cartesian product, called the composition operation
for each $a \in Obj(\mathcal{C})$ a point $id_a\in \mathcal{C}(a,a)$, called the identity morphism on $a$
such that the composition is associative and unital.
Given a topologically enriched category as in def. , then forgetting the topology on the hom-spaces (along the forgetful functor $U \colon Top_k \to Set$) yields an ordinary locally small category with
It is in this sense that $\mathcal{C}$ is a category with extra structure, and hence “enriched”.
The archetypical example is the following:
The category $Top_{cg}$ from def. itself, being a cartesian closed category, canonically obtains the structure of a topologically enriched category, def. , with hom-spaces given by compactly generated mapping spaces
and with composition
given by the (product$\dashv$ mapping-space)-adjunct of the evaluation morphism
A topologically enriched functor between two topologically enriched categories
is a $Top_{cg}$-enriched functor, hence:
a function
of objects;
for each $a,b \in Obj(\mathcal{C})$ a continuous function
of hom-spaces
such that this preserves composition and identity morphisms in the evident sense.
A homomorphism of topologically enriched functors
is a $Top_{cg}$-enriched natural transformation: for each $c \in Obj(\mathcal{C})$ a choice of morphism $\eta_c \in \mathcal{D}(F(c),G(c))$ such that for each pair of objects $c,d \in \mathcal{C}$ the two continuous functions
and
agree.
We write $[\mathcal{C}, \mathcal{D}]$ for the resulting category of topologically enriched functors. This itself naturally obtains the structure of topologically enriched category, see at enriched functor category.
For $\mathcal{C}$ any topologically enriched category, def. then a topologically enriched functor
to the archetypical topologically enriched category from example may be thought of as a topologically enriched copresheaf, at least if $\mathcal{C}$ is small (in that its class of objects is a proper set).
Hence the category of topologically enriched functors
according to def. may be thought of as the (co-)presheaf category over $\mathcal{C}$ in the realm of topologically enriched categories.
A functor $F \in [\mathcal{C}, Top_{cg}]$ is equivalently
a compactly generated topological space $F_a\in Top_{cg}$ for each object $a \in Obj(\mathcal{C})$;
for all pairs of objects $a,b \in Obj(\mathcal{C})$
such that composition is respected, in the evident sense.
For every object $c \in \mathcal{C}$, there is a topologically enriched representable functor, denoted $y(c)$ or $\mathcal{C}(c,-)$ which sends objects to
and whose action on morphisms is, under the above identification, just the composition operation in $\mathcal{C}$.
There is a full blown $Top_{cg}$-enriched Yoneda lemma. The following records a slightly simplified version.
(topologically enriched Yoneda-lemma)
Let $\mathcal{C}$ be a topologically enriched category, def. , write $[\mathcal{C}, Top_{cg}]$ for its category of topologically enriched (co-)presheaves, and for $c\in Obj(\mathcal{C})$ write $y(c) = \mathcal{C}(c,-) \in [\mathcal{C}, Top_k]$ for the topologically enriched functor that it represents, all according to example . Recall also the $Top_{cg}$-tensored functors $F \cdot X$ from that example.
For $c\in Obj(\mathcal{C})$, $X \in Top_{cg}$ and $F \in [\mathcal{C}, Top_{cg}]$, there is a natural bijection between
morphisms $y(c) \cdot X \longrightarrow F$ in $[\mathcal{C}, Top_{cg}]$;
morphisms $X \longrightarrow F(c)$ in $Top_{cg}$.
Given a morphism $\eta \colon y(c) \cdot X \longrightarrow F$ consider its component
and restrict that to the identity morphism $id_c \in \mathcal{C}(c,c)$ in the first argument
We claim that just this $\eta_c(id_c,-)$ already uniquely determines all components
of $\eta$, for all $d \in Obj(\mathcal{C})$: By definition of the transformation $\eta$ (def. ), the two functions
and
agree. This means that they may be thought of jointly as a function with values in commuting squares in $Top$ of this form:
For any $f \in \mathcal{C}(c,d)$, consider the restriction of
to $id_c \in \mathcal{C}(c,c)$, hence restricting the above commuting squares to
This shows that $\eta_d$ is fixed to be the function
and this is a continuous function since all the operations it is built from are continuous.
Conversely, given a continuous function $\alpha \colon X \longrightarrow F(c)$, define for each $d$ the function
Running the above analysis backwards shows that this determines a transformation $\eta \colon y(c)\times X \to F$.
With $Top_{cg}$ equipped with the classical model structure on topological spaces, which is a presentation for the archetypical (∞,1)-category ∞Grpd of ∞-groupoids, then the topological functor category
(def. , def. ) is a model for the (∞,1)-category of (∞,1)-presheaves on $\mathcal{C}^{op}$. This is made precise by the model structure on enriched functors, $[\mathcal{C},Top_{Quillen}]_{proj}$. See at classical model structure on topological spaces – Model structure on functors for details.
Last revised on December 20, 2023 at 09:34:07. See the history of this page for a list of all contributions to it.