The Topologist's sine curve, a useful example in point-set topology. It is connected but not path-connected.
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
The fundamental concepts in point-set topology are continuity, compactness, and connectedness:
Continuous functions, intuitively, take nearby points to nearby points.
Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
Connected sets are sets that cannot be divided into two pieces that are far apart.
The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.
Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.
In mathematics, generaltopology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions...
of topology, including differential topology, geometric topology, and algebraic topology. Another name for generaltopology is point-set topology. The...
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators,...
examples in generaltopology, a field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open...
natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which...
areas of topology, the focus here is on generaltopology. The following definitions are also fundamental to algebraic topology, differential topology and geometric...
of topological spaces in their own right is called point-set topology or generaltopology. Around 1735, Leonhard Euler discovered the formula V − E + F...
In mathematics, specifically generaltopology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean...
mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real...
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can...
In mathematics, and especially generaltopology, the Euclidean topology is the natural topology induced on n {\displaystyle n} -dimensional Euclidean space...
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else...
In topology and related branches of mathematics, a Hausdorff space (/ˈhaʊsdɔːrf/ HOWSS-dorf, /ˈhaʊzdɔːrf/ HOWZ-dorf), separated space or T2 space is a...
list of generaltopology topics. Topological space Topological property Open set, closed set Clopen set Closure (topology) Boundary (topology) Dense (topology)...
from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).[citation needed] Given a topological...
In generaltopology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X , {\displaystyle...
mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning...
distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental...
that a topology or topological space might possess; for that, see List of generaltopology topics and Topological property. Discrete topology − All subsets...
In generaltopology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology)...
{\displaystyle Y} , so that X ⊆ Y {\displaystyle X\subseteq Y} . In generaltopology, an embedding is a homeomorphism onto its image. More explicitly, an...
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly...
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on R {\displaystyle \mathbb {R} } , the set of real...
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