Theorem 3.1:

(i) If $X$ and $Y$ are topological spaces and $f : X \longrightarrow Y$ is a continuous map and $A$ is a connected subset of $X$ then $f(A)$ is a connected subset of $Y$.

(ii) A topological space $X$ is connected if and only if every continuous function $f : X \longrightarrow \mathbb{Z}$ is constant.

(iii) If $\{A_n\}$ is a sequence of connected subsets of a topological space $X$ and $${ A_n\cap A_{n+1}}$$ is non-empty for each $n = 1, 2, 3, \dots$ then $\cup_{n=1}^{\infty}A_n$ is connected. In particular, taking $A_2 = A_3 = \dots$ we get the result for two connected sets.

(iv) If $A_{\alpha}$ is a family of connected subsets of a topological space such that for some connected subset $B$, $A_{\alpha}\cap B \neq \emptyset$ for each $\alpha$, then $\bigcup_{\alpha} A_{\alpha}$ is also connected.

(v) Suppose that $A$ is a connected subset of a topological space and $A \subset B \subset \overline{A}$ then $B$ is also connected.

(vi) A space $X$ is connected if and only if the only subsets of $X$ that are open and closed are $X$ and $\emptyset$.