Concept

Excluded point topology

In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named: If X has two points, it is called the Sierpiński space. This case is somewhat special and is handled separately. If X is finite (with at least 3 points), the topology on X is called the finite excluded point topology If X is countably infinite, the topology on X is called the countable excluded point topology If X is uncountable, the topology on X is called the uncountable excluded point topology A generalization is the open extension topology; if has the discrete topology, then the open extension topology on is the excluded point topology. This topology is used to provide interesting examples and counterexamples. Let be a space with the excluded point topology with special point The space is compact, as the only neighborhood of is the whole space. The topology is an Alexandrov topology. The smallest neighborhood of is the whole space the smallest neighborhood of a point is the singleton These smallest neighborhoods are compact. Their closures are respectively and which are also compact. So the space is locally relatively compact (each point admits a local base of relatively compact neighborhoods) and locally compact in the sense that each point has a local base of compact neighborhoods. But points do not admit a local base of closed compact neighborhoods. The space is ultraconnected, as any nonempty closed set contains the point Therefore the space is also connected and path-connected.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Graph Chatbot

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.