In mathematics, a bouquet of circles or rose is a construction in topology occurring when some number of circles are "glued" to each other so that they share a single common point. The number of circles used in the construction may be finite or infinite. It is a special case of the more general wedge sum of pointed topological spaces. For other meanings of mathematics or math, see mathematics (disambiguation). ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In topology, the wedge sum is a one-point union of a family of topological spaces. ... In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ...

The covering space of figure-eight space. The lines labelled a and b represent moves on one or the other of the two generating circles.

The bouquet of two circles is known as figure-eight space. It has a fundamental group that is a free group in two generators. Thus, for example, the Cayley graph for two generators is the covering space for the figure eight space. Image File history File links Cayley_graph_of_F2. ... Image File history File links Cayley_graph_of_F2. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

More generally, a bouquet of n circles has a fundamental group that is the free group in n generators. However, this identification does not hold in the direct limit of . The bouquet of a countable infinity of circles is known as a Hawaiian earring; its fundamental group is not the free group in a countable infinity of generators. In mathematics, the Hawaiian earring is the topological space that arises by considering the one-point compactification of a countably infinite family of open intervals. ...