All of the Galois connections I know involving a power set arise from a relation $R : X \times Y \to 2$ as described for example here. As you observe, this relation can often be used to define a topological space. However, it can always be used to define a Chu space, which is in a precise sense a kind of generalized topological space.

Some definitions. The category $\text{Chu}$ of Chu spaces (in $\text{Set}$, over $2$) has as objects triplets $(X, Y, R)$ where $X, Y$ are sets and $R$ is a function $X \times Y \to 2$. A morphism $(X_1, Y_1, R_1) \to (X_2, Y_2, R_2)$ of Chu spaces is an adjunction: that is, it is a pair of functions $f_{\ast} : X_1 \to X_2$ and $f^{\ast} : Y_2 \to Y_1$ such that

$$R_2(f_{\ast}(x_1), y_2) = R_1(x_1, f^{\ast}(y_2)).$$

Many familiar categories turn out to be full subcategories of $\text{Chu}$.

Example. Suppose the map $Y \ni y \mapsto \{ x : xRy \forall y \in Y \} \in 2^X$ is injective. The full subcategory of Chu spaces with this property may be regarded as sets together with a distinguished collection of subsets, where $R$ is the incidence relation "the point $x \in X$ lies in the subset $S \in Y$," and where $f^{\ast}$ is consequently always the preimage along $f_{\ast}$, so the morphisms are precisely maps of sets such that the preimage of a distinguished subset is distinguished.

Subexample. In particular, topological spaces are such Chu spaces, and the notion of morphism agrees. Hence $\text{Top}$ is a full subcategory of Chu spaces.

This is the precise sense in which Chu spaces generalize topological spaces: being a topological space is a property of a Chu space, in the same way that being an abelian group is a property of a group.

Subexample. Similarly, the category $\text{Meas}$ of measurable spaces is a full subcategory of Chu spaces.

Example. Let $V$ be a vector space over $\mathbb{F}_2$. Then the dual pairing $\text{eval} : V \times V^{\ast} \to \mathbb{F}_2$ defines a Chu space. This inclusion is full.

Example. Similarly, let $B$ be a Boolean algebra with Stone space $S(B)$. Then the evaluation map $\text{eval} : B \times S(B) \to 2$ defines a Chu space. This inclusion is full.

Example. Let $P$ be a poset and let $2^P$ denote the poset of order-preserving functions $P \to 2$, where $2 = \{ 0, 1 \}$ and $0 < 1$. The evaluation map $\text{eval} : P \times 2^P \to 2$ defines a Chu space. This inclusion is full.

Example. $\text{Chu}$ is equipped with a natural involution switching $X$ and $Y$. Applying this involution to any of the above examples gives other examples. When applied to various subcategories of $\text{Chu}$ this involution reproduces many familiar dualities. For example, when applied to Boolean algebras, the involution reproduces Stone duality.

I haven't played much with Chu spaces, but my intuition is that one can think of $X$ as a set of "states" and $Y$ as a set of "observations" that one can make about states, with $R$ confirming whether or not a given observation is true of a given state. But then one can switch the roles of states and observations!

All of the Galois connections I know involving a power set arise from a relation $R : X \times Y \to 2$ as described for example here. As you observe, this relation can often be used to define a topological space. However, it can always be used to define a Chu space, which is in a precise sense a kind of generalized topological space.

Some definitions. The category $\text{Chu}$ of Chu spaces (in $\text{Set}$, over $2$) has as objects triplets $(X, Y, R)$ where $X, Y$ are sets and $R$ is a function $X \times Y \to 2$. A morphism $(X_1, Y_1, R_1) \to (X_2, Y_2, R_2)$ of Chu spaces is an adjunction: that is, it is a pair of functions $f_{\ast} : X_1 \to X_2$ and $f^{\ast} : Y_2 \to Y_1$ such that

$$R_2(f_{\ast}(x_1), y_2) = R_1(x_1, f^{\ast}(y_2)).$$

Many familiar categories turn out to be full subcategories of $\text{Chu}$.

Example. Suppose the map $Y \ni y \mapsto \{ x : xRy \forall y \in Y \} \in 2^X$ is injective. The full subcategory of Chu spaces with this property may be regarded as sets together with a distinguished collection of subsets, where $R$ is the incidence relation "the point $x \in X$ lies in the subset $S \in Y$," and where $f^{\ast}$ is consequently always the preimage along $f_{\ast}$, so the morphisms are precisely maps of sets such that the preimage of a distinguished subset is distinguished.

Subexample. In particular, topological spaces are such Chu spaces, and the notion of morphism agrees. Hence $\text{Top}$ is a full subcategory of Chu spaces.

This is the precise sense in which Chu spaces generalize topological spaces: being a topological space is a property of a Chu space, in the same way that being an abelian group is a property of a group.

Subexample. Similarly, the category $\text{Meas}$ of measurable spaces is a full subcategory of Chu spaces.

Example. Let $V$ be a vector space over $\mathbb{F}_2$. Then the dual pairing $\text{eval} : V \times V^{\ast} \to \mathbb{F}_2$ defines a Chu space. This inclusion is full.

Example. Similarly, let $B$ be a Boolean algebra with Stone space $S(B)$. Then the evaluation map $\text{eval} : B \times S(B) \to 2$ defines a Chu space. This inclusion is full.

Example. Let $P$ be a poset and let $2^P$ denote the poset of order-preserving functions $P \to 2$, where $2 = \{ 0, 1 \}$ and $0 < 1$. The evaluation map $\text{eval} : P \times 2^P \to 2$ defines a Chu space. This inclusion is full.

Example. $\text{Chu}$ is equipped with a natural involution switching $X$ and $Y$. Applying this involution to any of the above examples gives other examples. When applied to various subcategories of $\text{Chu}$ this involution reproduces many familiar dualities. For example, when applied to Boolean algebras, the involution reproduces Stone duality.

I haven't played much with Chu spaces, but my intuition is that one can think of $X$ as a set of "states" and $Y$ as a set of "observations" that one can make about states, with $R$ confirming whether or not a given observation is true of a given state. But then one can switch the roles of states and observations!

All of the Galois connections I know involving a power set arise from a relation $R : X \times Y \to 2$ as described for example here. As you observe, this relation can often be used to define a topological space. However, it can always be used to define a Chu space, which is in a precise sense a kind of generalized topological space.

Some definitions. The category $\text{Chu}(2)$ of Chu spaces over $2$ has as objects triplets $(X, Y, R)$ where $X, Y$ are sets and $R$ is a function $X \times Y \to 2$. A morphism $(X_1, Y_1, R_1) \to (X_2, Y_2, R_2)$ of Chu spaces is an adjunction: that is, it is a pair of functions $f_{\ast} : X_1 \to X_2$ and $f^{\ast} : Y_2 \to Y_1$ such that

$$R_2(f_{\ast}(x_1), y_2) = R_1(x_1, f^{\ast}(y_2)).$$

An astonishing number of other familiar categories turn out to be full subcategories of $\text{Chu}(2)$. Below I will abbreviate "Chu space over $2$" to "Chu space."

Example. Suppose the map $Y \ni y \mapsto \{ x : xRy \forall y \in Y \} \in 2^X$ is injective. The full subcategory of Chu spaces with this property may be regarded as sets together with a distinguished collection of subsets, where $R$ is the incidence relation "the point $x \in X$ lies in the subset $S \in Y$," and where $f^{\ast}$ is consequently always the preimage along $f_{\ast}$, so the morphisms are precisely maps of sets such that the preimage of a distinguished subset is distinguished.

Subexample. In particular, topological spaces are such Chu spaces, and the notion of morphism agrees. Hence $\text{Top}$ is a full subcategory of Chu spaces.

This is the precise sense in which Chu spaces generalize topological spaces: being a topological space is a property of a Chu space, in the same way that being an abelian group is a property of a group.

Subexample. Similarly, the category $\text{Meas}$ of measurable spaces is a full subcategory of Chu spaces.

Example. Let $V$ be a vector space over $\mathbb{F}_2$. Then the dual pairing $\text{eval} : V \times V^{\ast} \to \mathbb{F}_2$ defines a Chu space. This inclusion is full.

Example. Similarly, let $B$ be a Boolean algebra with Stone space $S(B)$. Then the evaluation map $\text{eval} : B \times S(B) \to 2$ defines a Chu space. This inclusion is full.

Example. Let $P$ be a poset and let $2^P$ denote the poset of order-preserving functions $P \to 2$, where $2 = \{ 0, 1 \}$ and $0 < 1$. The evaluation map $\text{eval} : P \times 2^P \to 2$ defines a Chu space. This inclusion is full.

Example. $\text{Chu}$ is equipped with a natural involution switching $X$ and $Y$. Applying this involution to any of the above examples gives other examples. When applied to various subcategories of $\text{Chu}$ this involution reproduces many familiar dualities. For example, when applied to Boolean algebras, the involution reproduces Stone duality.

I haven't played much with Chu spaces, but my intuition is that one can think of $X$ as a set of "states" and $Y$ as a set of "observations" that one can make about states, with $R$ confirming whether or not a given observation is true of a given state. But then one can switch the roles of states and observations!

All of the Galois connections I know involving a power set arise from a relation $R : X \times Y \to 2$ as described for example here. As you observe, this relation can often be used to define a topological space. However, it can always be used to define a Chu space, which is in a precise sense a kind of generalized topological space.

Some definitions. The category $\text{Chu}$ of Chu spaces (in $\text{Set}$, over $2$) has as objects triplets $(X, Y, R)$ where $X, Y$ are sets and $R$ is a function $X \times Y \to 2$. A morphism $(X_1, Y_1, R_1) \to (X_2, Y_2, R_2)$ of Chu spaces is an adjunction: that is, it is a pair of functions $f_{\ast} : X_1 \to X_2$ and $f^{\ast} : Y_2 \to Y_1$ such that

$$R_2(f_{\ast}(x_1), y_2) = R_1(x_1, f^{\ast}(y_2)).$$

Many familiar categories turn out to be full subcategories of $\text{Chu}$.

Example. Suppose the map $Y \ni y \mapsto \{ x : xRy \forall y \in Y \} \in 2^X$ is injective. The full subcategory of Chu spaces with this property may be regarded as sets together with a distinguished collection of subsets, where $R$ is the incidence relation "the point $x \in X$ lies in the subset $S \in Y$," and where $f^{\ast}$ is consequently always the preimage along $f_{\ast}$, so the morphisms are precisely maps of sets such that the preimage of a distinguished subset is distinguished.

Subexample. In particular, topological spaces are such Chu spaces, and the notion of morphism agrees. Hence $\text{Top}$ is a full subcategory of Chu spaces.

This is the precise sense in which Chu spaces generalize topological spaces: being a topological space is a property of a Chu space, in the same way that being an abelian group is a property of a group.

Subexample. Similarly, the category $\text{Meas}$ of measurable spaces is a full subcategory of Chu spaces.

Example. Let $V$ be a vector space over $\mathbb{F}_2$. Then the dual pairing $\text{eval} : V \times V^{\ast} \to \mathbb{F}_2$ defines a Chu space. This inclusion is full.

Example. Similarly, let $B$ be a Boolean algebra with Stone space $S(B)$. Then the evaluation map $\text{eval} : B \times S(B) \to 2$ defines a Chu space. This inclusion is full.

Example. Let $P$ be a poset and let $2^P$ denote the poset of order-preserving functions $P \to 2$, where $2 = \{ 0, 1 \}$ and $0 < 1$. The evaluation map $\text{eval} : P \times 2^P \to 2$ defines a Chu space. This inclusion is full.

Example. $\text{Chu}$ is equipped with a natural involution switching $X$ and $Y$. Applying this involution to any of the above examples gives other examples. When applied to various subcategories of $\text{Chu}$ this involution reproduces many familiar dualities. For example, when applied to Boolean algebras, the involution reproduces Stone duality.

I haven't played much with Chu spaces, but my intuition is that one can think of $X$ as a set of "states" and $Y$ as a set of "observations" that one can make about states, with $R$ confirming whether or not a given observation is true of a given state. But then one can switch the roles of states and observations!