##  [Local Connectedness](/local-connectedness-0) 

 Definition

A topological property requiring that every point has a neighborhood basis consisting entirely of connected open sets; equivalently, each point has arbitrarily small connected neighborhoods.

 

 

 

 

 

 





## Principle

Principle

Local connectedness controls the local topology: neighborhoods can be chosen connected, which implies components of open sets are open and simplifies the relation between components and path components; local connectedness is a local-to-global regularity hypothesis used in decomposition theorems.

 

 

 

 

 





## Demonstration

Demonstration

Example: Euclidean space R^n is locally connected because small open balls around any point are connected. A counterexample is the comb space where accumulation at a limit line produces points with no connected neighborhoods arbitrarily small, so the space fails to be locally connected.

 

 

 

 

## Misapplication

Misapplication

Assuming that connectedness of the whole space implies local connectedness, or that local connectedness implies path-connectedness; confusing local connectedness with the openness of components without checking the neighborhood basis condition.

 

 

 

 

 





## Consequence

Consequence

In a locally connected space, components of open sets are open, and in locally path-connected spaces path components agree with components; local connectedness often permits induction arguments and glueing procedures that rely on connected neighborhoods.

 

 

 

 

## Reversal

Reversal

Non-locally connected spaces have points with every sufficiently small neighborhood disconnected; such spaces can be connected globally while having wildly disconnected small-scale structure (e.g., certain fractals or comb-like examples).

 

 

 

 

 





## Boundary

Boundary

Property of topological spaces concerning bases of neighborhoods; not implied by mere connectedness and distinct from local path-connectedness (stronger). It applies to open neighborhood bases and does not assert global finiteness or compactness.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often contrasted with local path-connectedness: the latter requires path-connected neighborhoods and is stronger; local connectedness sits between global connectedness and stronger local path properties, producing tension when choosing hypotheses for theorems.

 

 

 

 

 





## Synthesis

Synthesis

Local Connectedness demands that each point admit a basis of connected open neighborhoods, a local regularity condition that makes components of open sets open and that eases passage from local information to global topological conclusions.