Here is an ad hoc attempt. A connectivity space is an ordered pair $\ \mathbf X:=(X\ \mathcal C)\ $ such that the following two axioms hold:

- $\ \left(A\ne\emptyset\ne B\,\ \wedge\ \,A\ B\in\mathcal C\right)\\ \quad \Rightarrow\quad \left( A\cup B\in \mathcal C\ \ \Leftrightarrow\quad\exists_{x\in A\cup B} \left(A\cup\{x\}\in\mathcal C\ \ \wedge\ \ B\cup\{x\}\in \mathcal C\right)\ \right) $
- $\ \forall_{x\ y\,\in\,A}\ \exists_{S\in\mathcal C}\ (x\ y\in S\ \ \wedge\ \ S\subseteq A)\quad\Rightarrow\quad A\in \mathcal C$

for every $\ A\ B\ \subseteq X.\ $ Next, given connectivity spaces $\ \mathbf X:=(X\ \mathcal C)\ $ and $\ \mathbf Y:=(Y\ \mathcal D),\ $ A connectivity map (or connectivity morphism) is any function $\ f:X\rightarrow Y\ $ such that $\ \forall_{A\in\mathcal C}\ f(A)\in\mathcal D.$

Given a topological space $\ \mathbf X:=(X\ T),\ $ we get the induced connectivity space $\ \mathbf X_c := (X\ \mathcal C_T),\ $ where $\ C_T\ $ is the family of all connected subsets of $\ \mathbf X$. Thus every continuous map between two topological spaces is a connectivity map between the induced connectivity spaces.

Do not expect that there is a very close relation between continuous maps and connectivity maps. But the relation between them should be interesting (a source of new MO-questions :-) ).

**REMARK** It follows from the above definition (two axioms) that $\ \emptyset\in\mathcal C.\ $ (*Thank you Eric for this point*).

**EXAMPLE** An intersection $\ \bigcap_{n=1}^\infty A_n\ $of a monotone sequence of closed connected subspaces $\ A_n\ $ doesn't have to be connected. For instance, consider the following subspaces of $\ \mathbb R^2$:

$$A_n\ :=\ \mathbb R^2\setminus (-1;1)\times(-n;n)$$

for $\ n=1\ 2\ \ldots$

Of course a small modification will give a similar example for open connected subsets $\ B_n,\ $ say:

$$B_n\ :=\ \mathbb R^2\setminus \{0\}\times[-n;n]$$

A point-free definition:

A connectivity structure $\ \mathcal C\ $ in set $\ X,\ $ with $\ \mathcal C_0:=\mathcal C\setminus \{\emptyset\},\ $ is defined by the following 3 axioms:

- $\forall_{A\ B\,\in\,\mathcal C_0}\ \left(A\cup B\in\mathcal C_0\quad\Leftrightarrow\quad\exists_{S\in\mathcal C_0}\left(S\subseteq A\cup B\ \ \wedge\ \ A\cup S\in \mathcal C_0\ \ \wedge\ \ B\cup S\in\mathcal C_0\right)\ \right)$
- $\forall_{R\ S\,\in\,\mathcal C_0}\, \left(\left(R\cup S\,\subseteq\, A\right)\ \Rightarrow\ \exists_{Q\in\mathcal C_0}\ R\cup S\subseteq Q\subseteq A\ \right)\quad\Rightarrow\quad A\in\mathcal C_0$
- $\emptyset\,\in\,\mathcal C$

for every $\ A\ B\,\in\,X$.

A categorical definition

**TERMINOLOGY** A category $\ \mathbf C\ $ is called *vague* $\ \Leftarrow:\Rightarrow\,\ \forall_{X\ Y\,\in\,Obj(\mathbf C)}\ |\,MOR(X\ Y)\,|\ \le\ 1$

**EXAMPLE** The category of all sets and of the **identity** embeddings is vague.

**DEFINITION 1 of a (connected) union** Let category $\ \mathbf C\ $ be vague. An object $\ C\ $ of objects $\ A\ B\ $ is called a *union* $\ \Leftarrow:\Rightarrow\ $ two conditions hold:

- $\ \exists_{S\,\in\,Obj(\mathbf C)}\\ \ \quad |\,MOR(S\ A)\,|\ =\ |\,MOR(S\ B)\,|\ =\ |\,MOR(A\ C)\,|\ =\ |\,MOR(B\ C\,|\ =\ 1$
- whenever $\ D\ $ is like $\ C\ $ above then $\ |\,MOR(C\ D)\,|\ =\ 1$.

Thus with every vague category $\ \mathbf C\ $we associate a u-graph, where two objects $\ A\ B\ $ are connected (i.e. form an edge of the connectivity graph) $\ \Leftarrow:\Rightarrow\ $ there exists a union of $\ A\ $ and $\ B.$

**DEFINITION 2 of merger** Let category $\ \mathbf C\ $ be vague. Objects $\ A\ B\ $ *merge* into an object $\ C\ \Leftarrow:\Rightarrow\ $ two conditions hold:

- $\ |\,MOR(A\ C)\,|\ =\ |\,MOR(B\ C)\,|\ =\ 1$
- whenever $\ D\ $ is like $\ C\ $ above, then $\ |\,|MOR(C\ D)\,|\ =\ 1$

With every vague category $\ \mathbf C\ $ we associate the merging graph of objects of $\ C\ $, where two objects form an edge $\ \Leftarrow:\Rightarrow\ $ if they merge.

**DEFINITION 3 of a connectivity category:**

A category $\ \mathbf C\ $ is a connectivity graph $\ \Leftarrow:\Rightarrow\ $ it satisfies the following four axioms:

$\ \mathbf C\ $ is vague;

every union of objects of $\ \mathbf C\ $ is a merger;

if $\ A\ B\ $ merge, and if $\ |\,MOR(A\ A')\,|\ 1\ $ then $\ A'\ $ and $\ B\ $ merge too;

- if $\ \mathbf D\ $ is a non-empty family of objects such that the full induced merger subgraph $\ \mathbf D\ $ is connected then there exists an object $\ C\ $ which is a merger of $\ \mathbf D,\ $ meaning that the following two conditions hold:
- $\ \forall_{A\in\mathbf D}\ \ |\,MOR(A\ C)\,|\ =\ 1$
- whenever $\ D\ $ is like $\ C\ $ then $\ \,MOR(C\ D)\,|\ =\ 1$

INTERPRETATION: The objects of a connectivity category play the role of non-empty connected spaces.

Back to topology:

In the case of a topological space $\ \mathbf X,\ $ the category $\ \mathbf C :=\mathbf C_{\mathbf X}\ $ consists of the non-empty connected subsets of $\ \mathbf X,\ $ and of the identity embeddings.