In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter for a point in a topological space is the collection of all neighbourhoods of
Neighbourhood of a point or set
An of a point (or subset) in a topological space is any open subset of that contains
A is any subset that contains open neighbourhood of ;
explicitly, is a neighbourhood of in if and only if there exists some open subset with .
Equivalently, a neighborhood of is any set that contains in its topological interior.
Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods."
Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a (respectively, , , etc.).
There are many other types of neighbourhoods that are used in topology and related fields like functional analysis.
The family of all neighbourhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open. Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.
Neighbourhood filter
The neighbourhood system for a point (or non-empty subset) is a filter called the The neighbourhood filter for a point is the same as the neighbourhood filter of the singleton set
A or (or or ) for a point is a filter base of the neighbourhood filter; this means that it is a subset
such that for all there exists some such that
That is, for any neighbourhood we can find a neighbourhood in the neighbourhood basis that is contained in
Equivalently, is a local basis at if and only if the neighbourhood filter can be recovered from in the sense that the following equality holds:
A family is a neighbourhood basis for if and only if is a cofinal subset of with respect to the partial order (importantly, this partial order is the superset relation and not the subset relation).
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.
Techniques et théories de base pour les équations aux dérivées partielles d'évolution. Etude d'exemples fondamentaux: équations du premier ordre, équation des ondes, équation de la chaleur. Théorème d
In mathematics, a filter on a set is a family of subsets such that: and if and , then If , and , then A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal.
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis. When a topology is generated using a family of pseudometrics, the space is called a gauge space.
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.
The social discourse surrounding the climate emergency progressively infuses the society, transforming into both micro- and macro-social injunctions to change. Yet, society - grounded in a territorial, social, and cultural contingency - appears to resist t ...
EPFL2023
Since the Industrial Revolution, urban spaces have undergone a profound reorganisation in parallel with a transformation of society and lifestyles. Urban sprawl has become widespread in Switzerland, as in countless countries, due to continuous innovations ...
With global environmental change and quality of life issues at the forefront of international discussions today, urban development strategies and policies are often framed around various sustainability or liveability goals. Regardless of the sustainability ...