Discover published sets by community

A subset of a topological space that contains all its limit points, and thus, its boundary. Example: The set of all points $x$ such that $0 \leq x \leq 1$ in the real number line is closed.

A topological space where for any two distinct points, there exist disjoint open sets containing each of the points. Example: The real number line with the usual topology is a Hausdorff space.

A function between topological spaces where the pre-image of every open set is open. Example: $f(x) = x^2$ is continuous in standard topology of real numbers.

A set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. Example: The real number line with the usual topology.

A surface with a single side and a single boundary component. Obtained by twisting a strip and connecting the ends. Example: A paper Möbius strip can be created by giving a paper strip a half-twist and joining the ends.

A topological space locally resembling Euclidean space near each point. Example: A 2D surface such as the surface of a sphere is a 2-dimensional manifold.

Any simple closed curve in the plane divides the plane into an inside and an outside region. Example: A circle on the plane separates the plane into an interior disk and an exterior unbounded region.

A property of a space indicating that every open cover has a finite subcover. Example: The closed interval $[0, 1]$ is compact in the real number line with the usual topology.

A topological space is connected if it cannot be divided into two disjoint nonempty open sets. Example: The real number line is connected, while the set of rationals is not.

The topology on a product of two topological spaces that generates open sets as products of open sets from each space. Example: The product topology on $\mathbb{R} \times \mathbb{R}$ is the standard Euclidean topology.

A study of mathematical knots, which are embeddings of circles into 3-dimensional spaces, up to isotopy. Example: The simplest non-trivial knot is the trefoil knot.

Comprises all loops in a space starting and ending at a base point, modulo homotopy. Example: The fundamental group of a circle is the set of integers, which represent winding numbers.

A bijective continuous function with a continuous inverse between two topological spaces. Example: Stretching a rubber band is homeomorphic as long as it does not tear.

A set of open sets in a topological space such that every open set can be written as a union of elements from the basis. Example: The set of all open intervals $(a,b)$ is a basis for the standard topology on the real numbers.

A space that maps onto another space such that each point in the latter has an open neighbourhood evenly covered by the mapping. Example: The real number line is a covering space for the unit circle via the exponential map.

A result in general topology stating that every complete metric space is of second category, meaning it cannot be the countable union of nowhere dense sets. Example: The real number line is a complete metric space and thus is of second category.

A type of topological space endowed with a metric, which measures the distance between points. Example: The set of real numbers with the function $d(x,y)=|x-y|$ is a metric space.

A continuous deformation of one function to another within a topological space. Example: A loop can be homotopically deformed to a point in Euclidean space.

Topology on a subset of a topological space where open sets are intersections of open sets from the larger space with the subset. Example: The set $[0,1]$ inherits the subspace topology from the real numbers.