📂Topology

Various Equivalent Conditions of a Connected Space

Definition ¹

For a topological space $X$, if a subset $A \subset X$ satisfies $$ A \ne \emptyset \\ A \ne X $$ then $A$ is called a proper Subset of $X$. For two proper subsets $A,B \subset X$, if $$ \overline{A} \cap B = \emptyset \\ A \cap \overline{B} = \emptyset $$ then $A$ and $B$ are called a separated set or simply a separation.

Equivalent Conditions of a Connected Space

Including the definition above, we can find various equivalent conditions of a connected space. First, just as we did with connected spaces, let us start with disconnected spaces.

Disconnected Space

The following propositions are equivalent to each other.

• (1): $X$ is a disconnected space.
• (2): $X$ is a union of some separated sets.
• (3): For the discrete space $\left\{ a, b \right\}$, there exists a surjective continuous function $f : X \to \left\{ a, b \right\}$.
• (4): There exists a proper subset that is both open and closed.
• (5): There exists a proper subset $A$ satisfying $\overline{A} \cap \overline{X \setminus A} = \emptyset$.

Connected Space

The following propositions are equivalent to each other.

• (1)’: $X$ is a connected space.
• (2)’: $X$ cannot be a union of any separated sets.
• (3)’: For the discrete space $\left\{ a, b \right\}$, there does not exist a surjective continuous function $f : X \to \left\{ a, b \right\}$.
• (4)’: There does not exist a proper subset that is both open and closed.
• (5)’: There does not exist a proper subset $A$ satisfying $\overline{A} \cap \overline{X \setminus A} = \emptyset$.

As you can see, all of them can be expressed as the negation of a disconnected space.