Publication

Types of Connectedness

Boris Buffoni
2025

Abstract

The differences between connectedness and path-connectedness, between connectedness and local connectedness, and between local connectedness at a point and weak local connectedness at a point are illustrated by examples, and it is shown that a metric space is locally connected at every point if and only if it is weakly locally connected at every point. It is also shown that if in addition such a metric space is connected and locally compact, it is path-connected. This result is key for all that follows.

Official source

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Boris Buffoni
2025

Abstract

Official source

Related concepts (32)

Locally connected space

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space.

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space is a if it is a connected space when viewed as a subspace of . Some related but stronger conditions are path connected, simply connected, and -connected.

Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

Related publications (3)

Types of Connectedness

Chat with Graph Search

Congestion Points

The Costly Path from Percolation to Full Connectivity

Browsing Recorded Meetings with Ferret