Components and local (path) connectedness

A topological space $X$ is called locally path connected at $x$ if for each open set $V$ containing $x$, there is a path connected open set $U$ with $x \in U \subset V$. It is called locally path connected if it is locally path connected at each $x \in X$. It is called locally connected at $x$ if, for each open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$. It is called locally connected if it is locally connected for each $x \in X$.

Show that a locally path connected space is locally connected.

Show that the topologist's sine curve (see the figure in Exercise 1.9.44) is not locally connected, where a space is locally connected at $x$ if for each open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$.

Show that an open set in $\mathbb {R}^{n}$ is locally path connected, where a space is locally path connected at $x$ if for each open set $V$ containing $x$, there is a path connected open set $U$ with $x \in U \subset V$.

Show that the path components of a locally path connected space are open sets, where a space is locally path connected at $x$ if for each open set $V$ containing $x$, there is a path connected open set $U$ with $x \in U \subset V$.

Show that if $X$ is locally path connected and connected, then $X$ is path connected.

Give an example of a path connected space which is not locally path connected.

Define an equivalence relation on $X$ by $x \sim y$ if there is a connected set containing both $x$ and $y$. The equivalence classes are called the components of $X$.

Prove that if an open set $U \subset \mathbb {R}^{2}$ is path connected, then any two points in $U$ can actually be connected by a polygonal path in $U$.

Show that a connected $n$-manifold is path connected.