"In mathematics, an open set is a generalization of an open interval in the real line."
Open sets are subsets of a space that are defined to be ‘open’ under some notion of openness. Closed sets, on the other hand, are defined to be complements of open sets, that is, a subset is closed if and only if its complement is open.
Topological Spaces: A topological space is a set with a collection of subsets, called open sets, that satisfy certain axioms.
Open Sets: A set is said to be open if every point in the set has a neighborhood contained entirely within the set.
Closed Sets: A set is said to be closed if it contains all its limit points.
Interior and Closure: The interior of a set is the largest open set contained in it, and the closure is the smallest closed set containing it.
Boundary: The boundary of a set is the set of points that are neither in the interior nor in the exterior.
Convergence: A sequence of points in a topological space is said to converge to a point if its limit is in the closure of the sequence.
Separation Axioms: Separation axioms are a set of axioms that describe how well-behaved a space is in terms of separating certain types of sets from each other.
Hausdorff Spaces: A Hausdorff space is a topological space in which any two distinct points have disjoint neighborhoods.
Compactness: A topological space is said to be compact if every open cover has a finite subcover.
Connectedness: A topological space is said to be connected if it is not the union of two disjoint non-empty open sets.
Open ball: An open subset of a metric space, defined by a point within the space and a positive radius around the point.
Closed ball: A closed subset of a metric space, defined by a point within the space and a non-negative radius around the point.
Interior: The largest open subset of a given set, containing only points from that set.
Closure: The smallest closed subset of a given set, containing all its limit points.
Boundary: The set of points that are neither in the interior nor in the exterior of a given set.
Complement: The set of points in a given metric space that do not belong to a given subset.
Convex: A subset of a metric space where the line segment between any two points in the subset is also within the subset.
Connected: A subset of a metric space that cannot be partitioned into two disjoint non-empty subsets.
Disjoint: Two subsets of a given metric space that do not have any common elements.
Dense: A subset of a given metric space where every point in the metric space is either in the set or is a limit point of the set.
"An open set is a set that, along with every point P, contains all points that are sufficiently near to P..."
"A set in which such a collection is given is called a topological space..."
"...the collection is called a topology. These conditions...allow enormous flexibility in the choice of open sets."
"For example, every subset can be open (the discrete topology)..."
"...no subset can be open except the space itself and the empty set (the indiscrete topology)."
"...open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces..."
"A topology allows defining properties such as continuity, connectedness, and compactness..."
"...manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space..."
"...no distance is defined in general."
"Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology..."
"...which is fundamental in algebraic geometry and scheme theory."
"...an open set is a generalization of an open interval in the real line."
"...open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces..."
"...every subset can be open (the discrete topology)..."
"...no subset can be open except the space itself and the empty set (the indiscrete topology)."
"These conditions...allow enormous flexibility in the choice of open sets."
"A topology allows defining properties such as continuity, connectedness, and compactness..."
"...topological spaces that, near each point, resemble an open set of a Euclidean space..."
"...fundamental in algebraic geometry and scheme theory."