Equivalent Definition for Locally Connected
Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space.
Then these definitions for local connectedness are logically equivalent:
- $T$ is locally connected iff the components of open subsets of $X$ are also open in $X$.
- $T$ is locally connected iff there is a base consisting entirely of connected sets.
Proof
The first equivalence is still unproven.
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Assume that $T$ is such that it has a basis $\mathcal B$ which consists entirely of connected sets.
For each $x \in X$ we define $\mathcal B_x = \left\{{B \in \mathcal B: x \in B}\right\}$.
This is a local basis.
As all the elements of $\mathcal B_x$ is also an element of $\mathcal B$, it follows that $\mathcal B_x$ is also formed of connected sets.
Thus, for each point $x \in X$, there is a local basis which consists entirely of connected sets.
Thus, $T$ is locally connected by definition.
$\Box$
Assume that $T$ is locally connected.
Then by definition, for each point $x \in X$, there exists a local basis $\mathcal D_x$ which consists entirely of open sets, each of which is connected.
Consider the set $\displaystyle \mathcal D = \bigcup_{x \in X} \mathcal D_x$.
From Union of Local Bases is Basis, $\mathcal D$ is a basis for the topology $\vartheta$.
Since each $\mathcal D_x$ consists entirely of connected sets, $\mathcal D$, the set of all this sets by definition, also consists entirely of connected sets.
$\blacksquare$
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 4$
