Existence of a Sub-Category of the Category of Topological Spaces

Question

My question start with the following observations:

1. If you have a finite number of topological spaces $X_1, \dots , X_n$ you can define a space that is the disjoint union of its $\sqcup_{i=1}^n X_n=Y$.

2. If you have a cardinal number $I$ and a family of topogical spaces $(X_i)_{i\in I}$ we can define the disjoint product of this spaces by $ \Pi X_i$

3. In this cases we can look that for any topological space in the first set i.e. $\{ X_1, \dots , X_n\}$ or $(X_i)_{i \in I}$ we can find a subspace $Z\subset Y$ such that the first space is isomorphic to $Z$ with the induced topology.

Question 1: Do exist a subcategory $S$ of the category of topological space such that there is a topological space $X$ in $S$ and for every topological space $Y$ in $S$ there exist a subset of $X$ that is isomorphic to $Y$ with the induced topology.

iv. If we look at the category of topological spaces it is false, and is false too for the category of Noetherian 0-dimensional topological spaces.

Question 2: Is possible to characterize the Sub-categories of the category of Topological spaces satisfying the following property: " There exist a topological space $X$ in $S$ such that for every topological space $Y$ in $S$ exist a subset of $X$ that is isomorphic to the space $Y$."

Answer 1

Spaces of the sort you are looking for are called universal.

Definition: A space $X$ is universal for a class of spaces $\mathcal{C}$ if it belongs to $\mathcal{C}$ and every space in $\mathcal{C}$ embeds into $X$.

Here are some examples:

1. Let $\Sigma = \lbrace{\bot, \top\rbrace}$ be the Sierpinski space, whose open subsests are $\emptyset$, $\lbrace\top\rbrace$ and $\Sigma$. The countable product $\Sigma^\omega$, of course equipped with the product topology, is a universal countably-based $T_0$-space. That is, every such space embeds in $\Sigma^\omega$. More generally, for any cardinal $\kappa$ the $\kappa$-fold product $\Sigma^\kappa$ is universal for $\kappa$-weighted $T_0$-spaces.

2. We can get rid of the $T_0$ condition by using instead of $\Sigma$ the space with three points, one of which is open, see "Universal Topological Spaces" by K. D. Magill, Jr. The American Mathematical Monthly , Vol. 95, No. 10 (Dec., 1988), pp. 942-946.

3. Urysohn universal space is universal for all separable metric spaces (even if we require an isometric embedding). This is by no means the only such space. A much simpler universal separable metric space is the space $C([0,1])$ of continuous real-valued maps, equipped with the compact-open topology.

4. A $0$-dimensional compact countably-based Hausdorff space embeds into Cantor space $2^\omega$, so that makes Cantor space universal for countably based Stone spaces. Again, we can go to higher weights by considering large products $2^\kappa$.

5. In domain theory there are universal domains (domains are certain kinds of posets with topologies and are important in theoretical computer science), and this has been studied quite systematically. In fact, there are general techchniques for constructing universal objects, see e.g., Droste, M., Göbel, R.: Universal Domains and the Amalgamation Property. Mathematical Structures in Computer Science (1993) 137-159. (Behind a paywall!)

6. The Hilbert cube is universal for countably based Tychonoff spaces.

One can also ask for spaces which are universal in a different way, namely spaces which cover all spaces of some class (rather than embed them). There are examples of these too:

1. Every inhabited complete separable metric space is a quotient of the Baire space, which is the countable product $\mathbb{N}^\omega$ of the discrete space of natural number $\mathbb{N}$.

2. If I am not mistaken (someone will correct me), the embedding of a countably based Stone space into Cantor space actually has a left inverse. So, every inhabited countably based Stone space is a quotient of Cantor space.

There are many other examples.

Answer 2

Let $\mathcal C\subset Top$ be the category of metrizable compact Hausdorff spaces.

There are two objects of $\mathcal C$ that are particularly remarkable:
the Cantor set $X:=\prod^\infty \{0,1\}$ and the Hilbert cube $Y:=\prod^\infty [0,1]$.

Every object of $\mathcal C$ (except $\emptyset$!) is a quotient space of $X$,
and every object of $\mathcal C$ is a subspace of $Y$.

Answer 3

Well, one subcategory satisfying the property you want is the subcategory of Graphs. There the object $X$ is the Rado graph (a.k.a. the Random Graph). All graphs occur as subgraphs of $X$. This is an example of a general phenomenon in logic known as a Fraisse Limit. The linked blog post also talks a bit about what's required in order to know a Fraisse limit exists, and that's as close to a characterization as I know. Also, Slide 29 here gives a very general characterization for when a subcategory of a category $C$ admits a Fraisse Limit.

Note, this article has a more hand-on and topological characterization for when a class of metric spaces admits a Fraisse Limit