The Australian Ballet is a premier ballet company based in Australia, known for its high-quality performances and contributions to the art of ballet. Founded in 1962, it has become one of the leading ballet companies in the Southern Hemisphere. The company is headquartered in Melbourne and performs at the Arts Centre Melbourne, among other venues across Australia. The Australian Ballet showcases a diverse repertoire that includes classical ballets, modern works, and contemporary pieces, often collaborating with renowned choreographers, composers, and artists.

In topology, theorems are statements that have been proven to be true based on axioms and previously established theorems within the framework of topology. Topology is a branch of mathematics that studies properties of space that are preserved under continuous transformations.

Formalism, in the context of deductive reasoning, refers to a method of logical reasoning that emphasizes the structure and form of arguments rather than their content or subject matter. In formalism, the validity of an argument is determined based on the logical relationships between premises and conclusions, which can be expressed through symbolic notation and formal systems.

Formalism, in philosophy, generally refers to a perspective that emphasizes the importance of form, structure, and rules in understanding and analyzing various phenomena, often in contrast to content or meaning. While formalism can be applied in various fields, including mathematics, logic, and art, its implications can vary depending on the context.

In mathematics, compactification is a technique used to extend a space such that it becomes compact. Compactness is a topological property that has important implications in various areas of mathematics, particularly in analysis and topology. ### General Idea The process of compactification typically involves adding "points at infinity" or otherwise altering the topology of a space to ensure that every open cover of the space has a finite subcover.

General topology, also known simply as topology, is a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. It provides the foundational language and concepts for many areas of mathematics by introducing notions such as continuity, compactness, connectedness, and convergence without relying on the traditional metrics of distance from geometry. Key concepts in general topology include: 1. **Topological Spaces**: A set equipped with a collection of open subsets that satisfy certain axioms.

Geometric topology is a branch of mathematics that studies the properties and structures of topological spaces that have a geometric nature. It merges concepts from both topology and geometry, focusing on the ways in which spaces can be shaped and how they can be manipulated. Here are some key aspects and areas of interest within geometric topology: 1. **Topological Spaces**: The study focuses on various types of topological spaces and their properties.

In group theory, the **center** of a group \( G \), often denoted as \( Z(G) \), is defined as the set of elements in \( G \) that commute with every other element in the group.

Low-dimensional topology is a branch of mathematics that deals with the properties and structures of topological spaces that are primarily in dimensions 2, 3, and sometimes 4. These dimensions are considered "low" in the sense that they are manageable and allow for a deeper, more intuitive understanding of topological phenomena compared to higher dimensions.

Molecular topology is a branch of chemistry that studies the spatial arrangement and connectivity of atoms within a molecule, focusing on the molecule's geometric and structural properties rather than its specific atomic identities or detailed configurations. It is a key aspect of theoretical and computational chemistry, as well as a tool in cheminformatics. The main concepts of molecular topology include: 1. **Graphs and Nodes**: Molecules can be represented as graphs, where atoms are nodes and chemical bonds are edges.

Topological spaces are fundamental objects of study in topology, a branch of mathematics focused on the properties of space that are preserved under continuous transformations. Here are some key properties and concepts associated with topological spaces: 1. **Open and Closed Sets**: - A topology on a set \(X\) is a collection of subsets of \(X\) (called open sets) that includes the empty set and \(X\) itself, and is closed under arbitrary unions and finite intersections.