The space $\mathbb{Q}$ is not connected. In fact, for any irrational $x \in \mathbb{R}$, the set $[-x,x] \cap \mathbb{Q}= (-x,x) \cap \mathbb{Q}$ is both open and closed in the subspace topology on $\mathbb{Q}$.

If $A$ is connected and $B$ is a set such that $A \subseteq B \subseteq \overline{A}$ then $B$ is also connected.

 Show that the complement of any countable subset $C$ in $\mathbb{R^2}$ is path-connected.

 Show that the continuous image of a path-connected space is path-connected.

For open subset $U$ of $\mathbb{R^n}$ , show that $U$ is connected if and only if $U$ is path-connected.

Let $X$ be a compact space with a nested sequence $\{C_n\}$ of non-empty closed subsets: $C_1 \supseteq C_2 \supseteq C_3 \cdots$ . Show that the intersection $\cap_nC_n$ is non-empty.

A topological space $X$ is compact if and only if for every collection $C$ with finite intersection property, $\cap_{C \in C}C$ is non-empty.

Let $X$ be a first countable space and let $A$ be a subset of $X$. Then the following are true:

(1) (Sequence Lemma) $A$ point $x \in \overline{A}$ if and only if there is a sequence of points of A converging to $x$.

(2) (Continuity Versus Sequential Continuity) Let $f: X \rightarrow Y$ . Then $f$ is continuous if and only if $f$ is sequentially continuous.

A metric space $(X,d)$ is second countable if any one of the following holds true:

(1) $X$ has a countable dense subset,

(2) $X$ is compact.

Show that $\mathbb{R_l}$ is not metrizable.

If $X$ is second countable then every open covering of $X$ contains a countable subcover.