Path Connected Topological Spaces Review

We will now review some of the recent content regarding path connected topological spaces.

• Recall from the Path Connected Topological Spaces page that a topological space $X$ is said to be Path Connected if for every pair of distinct points $x, y \in X$ there exists a continuous function $\alpha : [0, 1] \to X$ such that $\alpha(0) = x$ and $\alpha(1) = y$. The functions $\alpha$ are called Paths from $x$ and $y$.

• We saw that $\mathbb{R}^n$ with the usual topology is path connected and for any pair of points $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ we can define a path $\alpha : [0, 1] \to \mathbb{R}^n$ by:

(1)

\begin{align} \quad \alpha(t) = (1 - t) \mathbf{x} + t \mathbf{y} \end{align}

• On the Path Connectivity of Connected Topological Spaces page we saw that every path connected topological space is connected. The converse is not true in general. There exists some connected topological spaces that are not path connected. In this sense, path connectivity is a "stronger" type of connectivity.

• We then began to investigated the path connectivity of path connected sets. On the Path Connectivity of Countable Unions of Connected Sets we saw that if $\{ A_i \}_{i=1}^{\infty}$ is a countably infinite collection of path connected sets in a topological space $X$ and if further, $A_i \cap A_{i+1} \neq \emptyset$ for all $i \in I$ (i.e., successive sets overlap), then the resulting union $\displaystyle{\bigcup_{i=1}^{\infty} A_i}$ is also a path connected topological space.

• On the Path Connectivity of the Range of a Path Connected Set under a Continuous Function page we saw that if $X$ is a path connected topological space and $f : X \to Y$ is a continuous function then the range, $f(X)$ is a path connected topological space in $Y$.

• On the Path Connectedness of Arbitrary Topological Products page we saw that if $\{ A_i \}_{i \in I}$ is an arbitrary collection of path connected topological spaces then the topological product $\displaystyle{\prod_{i \in I} A_i}$ is also path connected. We proved this by first taking any two distinct points $(x_i)_{i \in I}$ and $(y_i)_{i \in I}$ in $\displaystyle{\prod_{i \in I} X_i}$ and then considering the projection maps defined for all $j \in I$ by $p_j((x_i)_{i \in I}) = x_j$. Since each of the spaces $X_j$ are path connected there exists continuos maps $\alpha_j : [0, 1] \to X_j$ such that $\alpha(0) = p_j((x_i)_{i \in I}) = x_j$ and such that $\alpha (1) = p_j((x_i)_{i \in I}) = y_j$, and so, we can define a function $\displaystyle{\alpha : [0, 1] \to \prod_{i \in I} X_i}$ for all $x \in [0, 1]$ by:

(2)

\begin{align} \quad \alpha(x) = (\alpha_j(x))_{j \in I} \end{align}

• Since each component of $\alpha$ is continuous we have that $\alpha$ is continuous which showed that $\displaystyle{\prod_{i \in I} X_i}$ is path connected.

• On the Path Connectedness of Open and Connected Sets in Euclidean Space page we looked at a very nice theorem regarding path connectivity of open and connected sets in Euclidean space $\mathbb{R}^n$. We saw that if $A$ is an open and connected set in $\mathbb{R}^n$ then $A$ is path connected.

• On the Locally Connected and Locally Path Connected Topological Spaces page we then began to define local connectivity and local path connectivity of a topological space.

• We said that a topological space $X$ is Locally Connected at $x$ if for every neighbourhood of $U$ of $x$ there exists a connected neighbourhood $V$ of $x$ with $x \in V \subseteq U$. Furthermore, we said that $X$ is locally connected (in general) if $X$ is locally connected at every point in $X$.

• Similarly, we said that a topological space $X$ is Locally Path Connected at $x$ if for every neighbourhood of $U$ of $x$ there exists a path connected neighbourhood $V$ of $x$ with $x \in V \subseteq U$. Furthermore, we said that $X$ is locally path connected (in general) if $X$ is locally path connected at every point in $X$.