# Topology: Handwritten Notes

A topological space is a collection of points with a topology-a structure that describes how close two points are to one another. It is a generalisation of Euclidean spaces that makes it possible to investigate boundaries, continuity, and connectivity. A topology is a group of open sets, or subsets, that adhere to certain principles.

A handwritten notes of Topology by Mr. Tahir Mehmood. These notes covers almost every topic which required to learn for MSc mathematics.

Name | Topology: Handwritten Notes |
---|---|

Author | Mr. Tahir Mahmood |

Pages | 262 pages |

Format | Scanned PDF |

Size | 10.08 mB |

### What is in the notes?

- Metric space
- Minkowski's inequality
- Open set
- Closed ball
- Closed set
- Bounded set
- Limit point
- Closure of a set
- Convergence in metric space and complete metric space
- Cauchy sequence
- Bounded sequence
- Nested interval property or Cantor's intersection theorem
- Continuous function
- Topological spaces
- Metric topology, cofinite topology
- Open set
- Closed set
- Closure of a set
- Neighbourhood
- Interior point, exterior point
- Boundary point
- Limit point (with respect to topology)
- Isolated point
- Dense
- Separable set; Countable set
- Base of topology
- Neighbourhood base or local base or base at a point
- Open cover; Lindelof space
- Lindelof theorem
- Relative topology, subspace
- Separation axioms; $T_0$-space
- $T_1$-space
- Subbase; Generation of topologies
- $T_2$-space
- Continuous function (with respect to topologies)
- Product topology
- Convergence of sequence in topological spaces

- Regular space
- Completely regular space
- Compactness in topological spaces
- Homeomorphism
- Countably compact space
- Bolzano Weierstrass property
- Lebesgue number; Big set; Lebesgue covery lemma
- $\varepsilon-$net; Totally bounded
- Connected spaces; Disconnected
- Component
- Totally disconnected
- Separated
- Normed spaced
- Uniformly continuous
- Closed unit ball; Convex set
- Vector space
- Linear combination; Spanning set; Linearly independent
- Linearly dependent
- Linearly independent lemma
- Finite dimensional; Subspace
- Equivalent norms
- Banach space
- Reiz Lemma
- Hilbert spaces; Inner product spaces
- Polarization identity
- Cauchy Schewarz inequality
- Appalonius identity
- Hilbert space; Pythagorian theorem
- Minimizing vector
- Direct sum
- Orthogonal set; Orthonormal set
- Bessel's inequality
- Total orthonormal sets (definition); Parsevel's equality
- Linear Operator; The Kernel or Null space of a linear operator; Continuous linear operator
- Bounded linear operator
- Norm of a bounded lienar operator
- Linear functionals

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