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Topology and Algebraic Topology
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Topology
Topology (from the Greek TOPO , "place", and LOGY, "study") is the mathematical study of shapes and spaces. It is a major area of mathematics concerned with the most basic properties of space, such as connectedness, continuity and boundary. It is the study of properties that are preserved under continuous deformations including stretching and bending, but not tearing or gluing. The exact mathematical definition is given below. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, referred to in Latin as the geometric situs ("geometry of place") or analysis situs (Greek-Latin for "picking apart of place"). This later acquired the name topology. By the middle of the 20th century, topology had become an important area of study within mathematics.
Definition:
Let $X$ be a set, and $\tau$ be a family of subsets of $X$ (i.e $\tau\subseteq \mathcal{P}(X)$ ). Then $\tau$ is called a topology of $X$ if
- Both the empty set and $X$ are elements of $\tau$,
- Any union of elements of $\tau$ is an element of $\tau$, and
- Any intersection of finitely many elements of $\tau$ is an element of $\tau$.
If $\tau$ is a topology on $X$ , then the pair $(X,\tau)$ is called a topological space. The elements of $\tau$ are called open sets of $X$,
Topology has many subfields.
- Point-set topology establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (examples include compactness and connectedness).
- Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
- Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots
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