Locally Connected Space

Definition

Locally Connected At a Point

Consider a Topological Space $(X,\mathcal{T})$ and a point $p\in X$. The Topological Space is locally connected at a point $p$ if every Open Set containing $p$ contains a connected Open Set which contains $p$. $\forall U\in \mathcal{T}\text{s.t.}U\ni p,\exists \text{connected}C\in \mathcal{T}\text{s.t.}p\in C\subset U$

Or, equivalently, the Topological Space is locally connected at a point $p$ if $p$ has a Local Basis consisting of open connected sets.

Locally Connected Space

Consider a Topological Space $(X,\mathcal{T})$ The Topological Space is locally connected If every point in $X$ is locally connected. $\forall p\in X,\forall U\in \mathcal{T}\text{s.t.}U\ni p,\exists \text{connected}C\in \mathcal{T}\text{s.t.}p\in C\subset U$

Or, equivalently, the Topological Space is locally connected if it has a basis consisting of open connected sets.

Examples

$C=(\{0\}\times [0,1])\cup (\{\frac{1}{n}|n\in \mathbb{N}\}\times [0,1])\cup ([0,1]\times \{0\})$ Topologist’s comb

The subspace $C$ in ${\mathbb{R}}^2$ is Connected Space but is not locally connected at $(0,t)$ where $0<t\le 1$.

The set of rational numbers $\mathbb{Q}$ in $\mathbb{R}$ is neither connected nor locally connected.

Facts

A Topological Space $(X,\mathcal{T})$ is locally connected if and only if for each open subset $U$ of the space $X$, each component $C_U$ of the subset $U$ is an Open Set. $\forall U\in \mathcal{T},C_U\in \mathcal{T}$