Connected Topological Spaces

Let $\left\{ C_{\lambda }\right\}_{\lambda \in \Lambda }$ be a collection of connected subsets of a topological space $X$ and suppose that any two of them have a point in common. Show that the union of $\left\{ C_{\lambda }\right\}_{\lambda \in \Lambda }$ also is connected.

Let $A$ be a connected subset of a topological space $X$, and suppose that $A \subseteq B \subseteq \bar{A}$. Show that $B$ is connected.

Show that the following subset of the plane is connected but not arcwise connected.

Show that an arcwise connected topological space $X$ is connected. Also show that each connected open subset $\mathcal{O}$ of a Euclidean space $\mathbb {R}^{n}$ is arcwise connected.

Consider the circle $C = \left\{ (x, y) \mid x^{2}+y^{2} = 1\right\}$ in the plane $\mathbb {R}^{2}$. Show that $C$ is connected.

Show that $\mathbb {R}^{n}$ is connected.

Show that a compact metric space ($X, \rho$) fails to be connected if and only if there are two disjoint, non-empty subsets $A$ and $B$ whose union is $X$ and $\epsilon > 0$ such that $\rho (u, v) \geq \epsilon$ for all $u \in A, v \in B$. Show that this is not necessarily the case for non-compact metric spaces.

Show that for any point ($x, y$) in the plane $\mathbb {R}^{2}$, the subspace $\mathbb {R}^{2} \sim \left\{ (x, y)\right\}$ is connected. Use this to show that $\mathbb {R}$ is not homeomorphic to $\mathbb {R}^{2}$.

Verify the equivalence of the three assertions in (4).