MTH634 Final Term Data

DATA for FINAL TERM ---- MTH634 ---- Module# 086 to 183

Module : 86-89

A map f: (X,TX)⟶ (Y,TY) is called a closed map iff the image of each closed subset of X is closed in Y. Projection map may or may not be closed. Is a continuous map an open map. (May or may not be Open) Is an open map a continuous map. (May or may not be Continuous) Restriction of an open map may or may not be an open map.

Module : 90-100

Homeomorphism: A map f: X ⟶ Y is called a Homeomorphism between two topological space (X,Tx) and (Y,Ty) iff 1. “ f ” is bijective. 2. “ f ” is continuous 3. F-1 : Y ⟶ X or “f” is open map Homeomorphic Space: Two topological space are called (X,Tx) and (Y,Ty) are called homeomorphic iff there exist a homeomorphism f: X ⟶ Y we also say such spaces are topologically equivalent spaces i.e . x ≅ y Equivalence Relation: A binary relation ~ on a set X is said to be equivalence relation iff ~ satisfied the

Homomorphism of topological space is an equivalence relation. A property “p” of a topological space (X,T) is called topological property if every space homeomorphic to X has property “p” i.e. A property that is unchanged under homeomorphism. Topological property is also called topological invariant. Some topological properties are: *_Cardinality of set X. *_Cardinality of T. *_Connectedness of space X. *_Being discrete. Some non-topological properties are: *_ Length *_ Area A metric is a distance function. A function that defines a distance between each pair of elements of a set.

Metric Space: A set X with a metric “d” defined on it is called a metric space.

Module : 101-107

𝑟} Prepare Page # 33-34-35 Metric Topology: Let X be a non empty set with metric d. The topology T on X generated by the set of all open balls in X with respect to d is called metric topology. A topology T on X induced by metric d. Consider X with a metric d, this d induces a topology “metric topology”. (x,d) is called metric space. The topology generated by the set of all singletons is discrete topology.

Module : 108-110

Metrizable Space: A topological space (X,T) is called Metrizable space if there exists a metric d on X that induces topology T. Not all spaces are Metrizable. Let X be an infinite set. Consider cofinite topology on X then it is not Metrizable space. ℝ with lower limit topology is not Metrizable space.

Module : 111-112

A topological space (X,T) is called a First Countable Space if for each 𝑝∈𝑋 there exists a countable local base at 𝑝. Every subspace of first countable space is first countable. Discrete space is first countable space. Every finite set with any topology is first countable.

Every metric space is first countable.

Module : 113-114

A topological space (X,T) is called Second Countable Space if there exist a countable a countable basis for T. Every subspace of a second countable space is second countable. Let X be a countable set with discrete topology then (X,Tdis) is second countable. Any finite set with any topology is second countable.

Module : 115-116

Second accountability implies first accountability. Let a topological space (X,T) is second countable space then it is also first countable space. First accountability does not imply second accountability. Consider ℝ with usual topology then it is both first countable and second countable. Consider ℝ with lower limit topology then it is first countable but not second countable. A first countable space may or may not be second countable.

Module : 117-119

Open Cover: Let (X,T) be topological space. An open cover of a X is a collection of subsets of X i.e.

Module : 120-121

Lindelof Space: A topological space (X,T) is called a Lindelof space iff every open cover of X has a countable sub cover. Let X be a finite set with any topology then it is Lindelof. A set with indiscrete topology then it is Lindelof. X= ℝ with usual topology is Lindelof. A subspace of Lindelof space needs not to be Lindelof. Every countable space is second Lindelof. (ALSO PREPARE ITS PROOF FROM P#69-70)

Module : 122-125

A topological space (X,T) is said to be separable if there exist a countable dense subset A of X. Every second countable space is separable. ℝ with lower limit topology is separable but not second countable. Let X be a finite set then X with any topology is separable. Consider a set X with indiscrete topology the X is separable. ℝ with usual topology is separable. ℝ with discrete topology is not separable. A discrete space is separable iff it is countable. [ALSO PROOF FROM PAGE # 74] X is separable iff X is countable. (X,Tcof) is separable. [ALSO PROOF FROM PAGE # 74-75] In general subspace of separable is not separable.

Module : 126

A metric space is not a second countable and not separable in general. An infinite set X with trivial metric is not second countable and not separable. A separable metric space is second countable. [ALSO PROOF FROM PAGE # 78]

Module : 127-143

There are 7 separation axioms. T0-Spaces, T1-Spaces, T2-Spaces (Hausdorff), T3-Spaces, Normal Spaces,T4Spaces,

T1-Propery is a topology property. Every discrete space is T1 space. Every metric space is T1 space. Every cofinite space is T1 space. T1 property is much stronger property than T0 property. A topological space (X,T) is T1 iff every singleton subset {𝑥} ⊂𝑋 is closed.[ALSO PROOF FROM PAGE #