## MTH634: Topology

The subject of topology discusses those geometric mathematical structures that are not altered by topological mappings. MTH634 Handouts pdf

# MTH634 Handouts pdf

Course Category:  MTH634 Handouts pdf

#### Course Outline

Topological Spaces: Definition and Examples, Open Sets, Closed Sets,  Basis for a Topology, Subbasis, Comparable Topologies, Topologies on Real Line, Subspace Topology, Relative Open Sets, Interior Points, Interior of a Set, Limit Points, Closure of a Set, Exterior, Boundary of a Set, First and Second Countable Space, Separable Spaces,  Metric Topology, Continuous Functions, Open Maps, Closed Maps, Homeomorphic Spaces, Topological Properties, Separation Axioms, Hausdorff Spaces, Regular Spaces, Normal Spaces, Urysohn’s Lemma, Compact Spaces, Connected Spaces, Connected Components, Locally Connected spaces, Path Connected Spaces, MTH634 Handouts pdf

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## MTH634: Topology

Topology studies the structures of fixed spaces under any continuous flexibility. It is sometimes called “rubber-sheet geometry” because objects can be expanded and shortened like rubber, but they cannot be broken. For example, a square can be broken into a circle without breaking it, but figure 8 cannot. The square is therefore the same as the weather and the circle, but different from Figure 8.

#### The Main Problem of Topology

The subject of topology discusses those geometric mathematical structures that are not altered by topological mappings, that is, by bijective maps (i.e., one-to-one interactions) and continuous (i.e., continuous, continuous crossings). A geometric image is understood as a point placed in a 3-dimensional space (or in a space of higher magnitude). The continuous map is a map of the progressive activities in the Cartesian communication system of this space. Map operations need to be defined only by image points, and do not need to be defined throughout the space. Those structures that remain unchanged under a topological map are called topological structures of mathematics.

Two figures that can be mapped in a natural setting are said to be homeomorphic. For example, the surface of the hemisphere and the circular disk are homeomorphic, because one can map the hemisphere topologically to the disc using orthogonal projection. Generally, any two areas that can be disabled by each other, by bending and twisting, are homeomorphic. For example, the top of the circle, the top of the cube, and the ellipsoidal surface are homeomorphic; the annulus and the cylindrical surfaces of homeomorphic average length.

#### Isotop and homotopy

##### Isotropic

We first consider curves that do not have double points and have a straightforward sense of cut, i.e., topological images of directed circles. One obvious phase of all such curves is closed to consider the two curves a and b to be equal if each can be continuously deformed on the other at the top. First, we look at isotopic modification. This aging causes curve a to remain duplicated in all its central points during its transition to b. In this case, a and b are called isotopic

##### Homotropic

In homotopic deformation of a to b it is not necessary for the curve to remain without double points in all centered areas. Instead, a may contradict itself during its conversion to b. If a and b, we no longer think they have no double points, can be converted to another by homotopic fluctuations they are said to be homotopic, or more precisely, freely homotopic. Isotopic curves are already homotopic.