My question start with the following observations:

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 $\sqcup_{i=1}^n X_n=Y$.

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$

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 $Y$."