Topology Qualifying Syllabus - Revision history

Kfagerstrom: /* Homeomorphism invariants: */

2023-05-21T22:02:58Z

Homeomorphism invariants:

Kfagerstrom: /* Topological spaces and continuous functions: */

2023-05-21T21:06:33Z

Topological spaces and continuous functions:

Kfagerstrom: /* Point-set Topology: */

2023-05-21T21:05:53Z

Point-set Topology:

Kfagerstrom: /* Homeomorphism invariants: */

2023-05-21T20:50:01Z

Homeomorphism invariants:

Kfagerstrom: Created page with "==Point-set Topology:== ===Topological spaces and continuous functions:=== Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit..."

2023-05-21T20:46:58Z

Created page with "==Point-set Topology:== ===Topological spaces and continuous functions:=== Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit..."

New page

==Point-set Topology:==
===Topological spaces and continuous functions:===
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.
===Homeomorphism invariants:===
Separation properties (T0, T1, Hausdorff, regular, normal),
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.
===Continuous deformations:===
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.
==Algebraic topology:==
===Fundamental groups:===
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
===Covering spaces:===
Covering map, Lifting theorems; covering space group action; universal
covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal
covering; applications to group theory.
===Homology:===
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.

← Older revision		Revision as of 22:02, 21 May 2023
Line 10:		Line 10:

	[[Path Connected]]		[[Path Connected]]
		+	[[ T2 (Hausdorff)]]
		+	[[Compact]]

	===Continuous deformations:===		===Continuous deformations:===

@@ Line 3: / Line 3: @@
 Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.
-[[Quotient Topology]]
+[[Quotients (Topology)]]
 ===Homeomorphism invariants:===

@@ Line 2: / Line 2: @@
 ===Topological spaces and continuous functions:===
 Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.
 ===Homeomorphism invariants:===
 Separation properties (T0, T1, Hausdorff, regular, normal),
 countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.
 [[Path Connected]]
 ===Continuous deformations:===
 Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.
 ==Algebraic topology:==
 ===Fundamental groups:===