see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
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
Basic homotopy theory
In a (∞,1)-category $C$ admitting a final object ${*}$, for any object $X$ its suspension object $\Sigma X$ is the homotopy pushout
This is the mapping cone of the terminal map $X \to {*}$. See there for more details.
This concept is dual to that of loop space object.
Let $C$ be a pointed (infinity,1)-category. Write $M^\Sigma$ for the (infinity,1)-category of cocartesian squares of the form
where $X$ and $Y$ are objects of $C$. Supposing that $C$ admits cofibres of all morphisms, then one sees that the functor $M^\Sigma \to C$ given by evaluation at the initial vertex ($X$) is a trivial fibration. Hence it admits a section $s : C \to M^\Sigma$. Then the suspension functor $\Sigma_C : C \to C$ is the composite of $s$ with the functor $M^\Sigma \to C$ given by evaluating at the final vertex ($Y$).
$\Sigma_C$ is left adjoint to the loop space functor $\Omega_C$.
For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, the suspension object $\Sigma X$ is homotopy equivalent to $B{\mathbb{Z}}\wedge X$, the smash product by the classifing space of the discrete group of integers.
We outline a proof below. For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, its reduced free group, denoted by $F[X]$, is the left adjoint to the functor $\Omega {\mathbf{B}}:Grp(\mathcal{H})\to \mathcal{H}_*$ which sends a group object internal to ${\mathcal{H}}$ to the loop space of its delooping object.
For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence ${\mathbf{B}}F[X]\simeq \Sigma X$.
This is due to the adjunction $(\Sigma \vdash \Omega):\mathcal{H}_*\leftrightarrows\mathcal{H}_*$ between suspending and looping and the the adjunction $(\Omega \vdash {\mathbf{B}}):PathConn(\mathcal{H}_*)\leftrightarrows Grp(\mathcal{H})$ between looping and delooping. Indeed, for any group object $H$, the above-mentioned adjunctions imply the following natural equivalences:
Hence $\Omega \Sigma X$ has the universal property of the reduced free group. Delooping gives the required result.
The (∞,1)-category $Grp(\mathcal{H})$ of group objects internal ${\mathcal{H}}$ is tensored over ${\mathcal{H}}_*$; in particular, for $G$ a group object and $X$ a pointed object, we can form the tensor product $X\otimes G$, which is a group object. Explicitly, this tensor product is required to satisfy a homotopy equivalence $Grp({\mathcal{H}})(\Omega (X\otimes G, H)\simeq PathConn({\mathcal{H}}_*)(X, Grp({\mathcal{H}})(G, H))$, natural in group objects $H$.
For $X$ a pointed object and $G$ a group object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence ${\mathbf{B}}(X\otimes G)\simeq X\wedge {\mathbf{B}}G$.
This is due to the adjunction $(\Omega \vdash {\mathbf{B}}):PathConn(\mathcal{H}_*)\leftrightarrows Grp(\mathcal{H})$ between looping and delooping and the internal hom adjunction. Indeed, for any group object $H$, the above-mentioned adjunctions gives the following natural equivalences:
Hence $\Omega (X\wedge {\mathbf{B}}G)$ has the universal property of the tensor product. Delooping gives the required result.
For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence $F[X]\simeq X\otimes Z$, where $Z$ is the group object whose delooping object is $B {\mathbb{Z}}$, the classifying space of the discrete group of integers.
Since ${\mathcal{H}}$ is a Grothedieck $(\infty,1)$-topos, the $(\infty,1)$-functor $*\to {\mathbf{B}}-:Group(\mathcal{H})\to Func(\Delta^1,\mathcal{H})$ which sends a group object to the map from the terminal object to its delooping object is a $(\infty,1)$-categorial equivalence onto its image, which is the full subcategory of $Func(\Delta^1,\mathcal{H})$ spanned by the effective epimorphisms from the terminal object. Hence, for $H$ a group object, we have
This latter based mapping object is equivalent to the based object of deloopable maps from ${\mathbb{Z}}$ to $\Omega{\mathbf{B}}H$, which is just $\Omega{\mathbf{B}}H$, since ${\mathbb{Z}}$ is the discrete free group on one generator.
Hence, there are the following natural equivalences:
Therefore $F[X]$ has the universal property of the tensor product $X\otimes Z$. The required natural equivalence follows by abstract nonsense.
For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence $\Sigma X\simeq B{\mathbb{Z}}\wedge X$.
Deloop the natural equivalence in Lemma 1 to obtain the natural equivalence ${\mathbf{B}}F[X]\simeq {\mathbf{B}}(X\otimes Z)$. By propositions 1 and 2, this gives the required natural equivalence.
Let $C$ be a category admitting small colimits. Let $\Phi$ be a graded monoid in the category of groups and $F : C \to C$ a $\Phi$-symmetric endofunctor of $C$ that commutes with small colimits. Let $Spect_F^{\Phi}(C)$ denote the category of $\Phi$-symmetric $F$-spectrum objects in $C$.
Following Ayoub, the evaluation functor
which “evaluates” a symmetric spectrum at its $n$th component, admits under these assumptions a left adjoint
called the $n$th suspension functor, more commonly denoted $\Sigma_C^{\infty-n}$.
When $C$ is symmetric monoidal, and in the case $\Phi = \Sigma$ and $F = T \otimes -$ for some object $T$, there is an induced symmetric monoidal structure on $Spect^\Sigma_T(C)$ as described at symmetric monoidal structure on spectrum objects.
Proposition. One has
for all $X,Y \in C$. In particular, $Sus = Sus^0 : C \to \Spect^\Sigma_T(C)$ is a symmetric monoidal functor.
In Top, this is the reduced suspension of a space.
In a category of chain complexes the suspension of a chain complex is given by shifting the degrees of the chain complex up by one.
suspension object
A detailed treatment of the 1-categorical case is in the last chapter of