Local Connectivity and Local Path Connectivity
Definition
Let us call $X$ a topological space.
- If for all $U$ containing $x \in X$, there exists an open connected set $C$ that satisfies $x \in C \subset U$, then $X$ is said to be locally connected at $x$. If it is locally connected for all $x \in X$, then $X$ is termed a locally connected space.
- If for all $U$ containing $x \in X$, there exists an open path-connected set $P$ that satisfies $x \in P \subset U$, then $X$ is said to be locally path-connected at $x$. If it is locally path-connected for all $x \in X$, then $X$ is termed a locally path-connected space.
Theorem
Local connectivity
- [1-1]: The necessary and sufficient condition for $X$ to be locally connected at $x$ is for $X$ to have a local basis that includes open connected sets.
- [1-2]: The necessary and sufficient condition for $X$ to be a locally connected space is for $X$ to have a basis that includes open connected sets.
- [1-3]: The necessary and sufficient condition for $X$ to be a locally connected space is for every connected component of open $O \subset X$ to be an open set in $X$.
- [1-4]: Local connectivity is a topological property.
- [1-5]: Local connectivity and connectivity do not have an inclusion relationship.
Local path-connectivity
- [2-1]: The necessary and sufficient condition for $X$ to be locally path-connected at $x$ is for $X$ to have a local basis that includes open path-connected sets.
- [2-2]: The necessary and sufficient condition for $X$ to be a locally path-connected space is for $X$ to have a basis that includes open path-connected sets.
- [2-3]: The necessary and sufficient condition for $X$ to be a locally path-connected space is for every path-connected component of open $O \subset X$ to be an open set in $X$.
- [2-4]: Local path-connectivity is a topological property.
- [2-5]: Local path-connectivity and path-connectivity do not have an inclusion relationship.
Relationship between local connectivity and local path-connectivity
- [3]: A locally path-connected space is a locally connected space.
- [4]: A space that is both connected and locally path-connected is a path-connected space.
Explanation
Even before reading, it may seem tedious and the language complicated, but the concept itself is not that significant. The reason for methodically writing out so much information is because loosely accepting these concepts through mathematical intuition often results in misconceptions.
Local connectivity and connectivity, local path-connectivity and path-connectivity are completely separate concepts. As mentioned in theorem [3], it’s really, luckily, that a locally path-connected space being a locally connected space.
Example of [1-5]
It suffices to show a counterexample to the claim that local connectivity and connectivity have an inclusion relationship.
The topologist’s sine curve is a connected space but not a locally connected space.
A discrete space is a locally connected space but not a connected space hence it is not a connected space.
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Example of [2-5]
It suffices to show a counterexample to the claim that local path-connectivity and path-connectivity have an inclusion relationship.
The topologist’s comb space is a path-connected space but not a locally path-connected space.
A discrete space is a locally path-connected space but not a connected space hence it is not a path-connected space.
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