[Edit: Corrected some false claims and modified questions accordingly.]

Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point.
This is conventionally known as the $(\infty, 1)$-category of spaces, but for the purposes of this question it is probably best to avoid that terminology.
A lot is known about $\mathcal{S}$, but my question is about a different $(\infty, 1)$-category – the one you would get if one naïvely attempted to form the $(\infty, 1)$-category of topological spaces.

Let $\textbf{Top}$ be the category of _all_ topological spaces.
I mean to include the bad ones – topological spaces not locally contractible, topological spaces not having the homotopy type of a CW-complex, non-discrete totally disconnected spaces, non-$T_0$ spaces, you name it.
As is well known, $\textbf{Top}$ is _not_ cartesian closed, but this is no obstruction to getting a decent simplicial enrichment:

1. Given topological spaces $X$ and $Y$, we have a natural simplicial set $\textrm{Map} (X, Y)$ where the $n$-simplices are the continuous maps $X \times \left| \Delta^n \right| \to Y$.
From this definition it is immediate that $\textrm{Map} (X, -) : \textbf{Top} \to \textbf{sSet}$ preserves limits and therefore has a left adjoint, which I write as $X \odot {-} : \textbf{sSet} \to \textbf{Top}$.

2. Since $\left| \Delta^n \right|$ is compact Hausdorff, it is exponentiable a fortiori, so $\textrm{Map} (X, Y)_n$ is equivalently the set of continuous maps $X \to Y^{\left| \Delta^n \right|}$.
Hence $\textrm{Map} (-, Y) : \textbf{Top}^\textrm{op} \to \textbf{sSet}$ also preserves limits and has a left adjoint, which I write as ${-} \pitchfork Y : \textbf{sSet} \to \textbf{Top}^\textrm{op}$.

   (This probably surprises some people, since $\left| K \right|$ is not necessarily exponentiable for an infinite simplicial set $K$.
The short answer is that $X \odot K$ is not necessarily the same as $X \times \left| K \right|$, and $K \pitchfork Y$ is not necessarily the same as $Y^{\left| K \right|}$.
Rather, $X \odot K = \int^n X \times \left| \Delta^n \right| \times K_n$ and $K \pitchfork Y = \int_n Y^{\left| \Delta^n \right| \times K_n}$.
When $X$ (!!) is exponentiable then $X \odot K \cong X \times \left| K \right|$.)

3. There is an evident simplicial composition map $\textrm{Map} (Y, Z) \times \textrm{Map} (X, Y) \to \textrm{Map} (X, Z)$: given $g : Y \times \left| \Delta^n \right| \to Z$ and $f : X \times \left| \Delta^n \right| \to Y$, define $g \circ_n f$ to be the composite
$$\require{AMScd}
\begin{CD}
X \times \left| \Delta^n \right| 
@>{\textrm{id}_X \times \delta_{\left| \Delta^n \right|}}>>
X \times \left| \Delta^n \right| \times \left| \Delta^n \right|
@>{f \times \textrm{id}_{\left| \Delta^n \right|}}>> Y \times \left| \Delta^n \right| @>{g}>> Z
\end{CD}$$
where $\delta_{\left| \Delta^n \right|} : \left| \Delta^n \right| \to \left| \Delta^n \right| \times \left| \Delta^n \right|$ is the diagonal embedding.

   (Basically we are exploiting the fact that every object – so $\left| \Delta^n \right|$ in particular – in a cartesian monoidal category has a unique comonoid structure.
You can do this kind of thing whenever you have a comonoid in a monoidal category.)

4. We get an $\textbf{sSet}$-enrichment of $\textbf{Top}$, but unfortunately it is not clear whether it it tensored and cotensored.
I have seen it claimed that we have tensors and cotensors for finite simplicial sets, but there is some point-set topology involved in checking this.
I think finite tensors and cotensors suffice, at any rate.

   (If we restrict to a complete and cocomplete cartesian closed subcategory of $\textbf{Top}$ containing the standard simplices there is no problem, and the subtlety about $\odot$ and $\pitchfork$ also goes away.)

5. In fact, $\textrm{Map} (X, Y)$ is a Kan complex for all $X$ and $Y$: indeed, the natural map
$$\textrm{Hom}_\textbf{sSet} (\Delta^n, \textrm{Map} (X, Y)) \to \textrm{Hom}_\textbf{sSet} (\Lambda^n_k, \textrm{Map} (X, Y))$$
is a split surjection, because it can be naturally identified with the natural map
$$\textrm{Hom}_\textbf{Top} (X, Y^{\left| \Delta^n \right|}) \to \textrm{Hom}_\textbf{Top} (X, Y^{\left| \Lambda^n_k \right|})$$
induced by the inclusion $\left| \Lambda^n_k \right| \to \left| \Delta^n \right|$, which has a (non-canonical) retraction.

Thus, we have a complete and cocomplete $(\infty, 1)$-category $\mathcal{T}$ in the style of 1970–80s homotopy theory, and if you insist, we can apply the homotopy coherent nerve functor to get a quasicategory, or take the nerve to get a Segal space, or whatever you please.

**Observation 1.**
Two continuous maps $X \to Y$, considered as vertices of $\textrm{Map} (X, Y)$, are in the same connected component if and only if they are homotopic in the classical sense.
In fact, an edge of $\textrm{Map} (X, Y)$ is literally a homotopy between continuous maps $X \to Y$ in the classical sense.
So the homotopy category of $\mathcal{T}$ is the _classical_ homotopy category, which we know to strictly contain the homotopy category of $\mathcal{S}$: we can model $\mathcal{S}$ as the full (simplicially enriched) subcategory $\textbf{CW} \subset \textbf{Top}$ consisting of the CW-complexes, and we know that there are topological spaces not homotopy equivalent to any CW-complex.

**Question 1.**
If I understand correctly, Strøm showed that there is a model structure on $\textbf{Top}$ where the "weak equivalences" are the classical homotopy equivalences.
Thus, by general theory, we can construct a complete and complete $(\infty, 1)$-category by localising the ordinary 1-category $\textbf{Top}$ with respect to homotopy equivalences.
Does $\mathcal{T}$, as I defined it above, coincide with what general theory gives us?

**Observation 2.**
$\mathcal{S}$ is a reflective localisation of $\mathcal{T}$.
This would be a consequence of general theory (because the Quillen model structure is a left Bousfield localisation of the Hurewicz model structure) if the answer to question 1 is yes, but I think the usual construction of the Quillen model structure already contains a direct proof of this fact.

**Question 2.**
Does the reflector $\mathcal{T} \to \mathcal{S}$ preserve finite products?
What about finite limits in general?

**Observation 3.**
The functor ${-} \times T : \textbf{Top} \to \textbf{Top}$ very nearly preserves all colimits: in fact, it preserves all colimits at the level of point sets, so the only thing that can go wrong is the topology on the colimit.
If it preserved all colimits it would automatically have a right adjoint, making $T$ an exponentiable space.
Since ${-} \times T$ preserves coproducts, the failure of $\textbf{Top}$ to be cartesian closed is down to the failure of ${-} \times T$ to preserve quotient maps (= regular/strong/extremal epimorphisms) when $T$ is not a nice topological space.
But I think the situation is much better for homotopy colimits: ${-} \times T$ preserves subspace inclusions (= regular/strong/extremal monomorphisms), hence it preserves pushouts of subspace inclusions as well as directed colimits of subspace inclusions, and I believe this implies ${-} \times T$ preserves all homotopy colimits.

**Question 3.**
Does ${-} \times T$ have a homotopy right adjoint?
In other words, is $\mathcal{T}$ a cartesian closed $(\infty, 1)$-category?
In $\textbf{Top}$ the special adjoint functor theorem is available and implies $\textbf{Top}$ is a total category, but it is not clear to me whether $\mathcal{T}$ is similarly blessed.
(My understanding is that there is no analogue of the SAFT in $(\infty, 1)$-category theory.)

**Observation 4.**
$\textbf{Top}$ fails to be a locally presentable category for two reasons:

* $\textbf{Top}$ does not have a small dense generating set of objects, i.e. there is no small full subcategory $\mathbf{C}$ such that the Yoneda representation $\textbf{Top} \to [\mathbf{C}^\textrm{op}, \textbf{Set}]$ is fully faithful.

* There are objects in $\textbf{Top}$ "without rank", i.e. there is a topological space $X$ such that $\textrm{Hom}_\textbf{Top} (X, {-}) : \textbf{Top} \to \textbf{Set}$ does not preserve $\lambda$-filtered colimits for any $\lambda$.

**Question 4.**
I recall hearing that $\mathcal{T}$ is not a locally presentable $(\infty, 1)$-category.
Why not?
Is there still no small dense generating set of objects?
(This seems likely to me.)
Are there still objects "without rank"?
(I have no intuition about this, even for $\textbf{Top}$.)

I imagine some of these questions have well-known answers, but it would be nice to a definitive resource for novices confused/curious about the difference between $\mathcal{S}$ and $\mathcal{T}$.