# Category:Topology

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This category contains results about Topology.

Definitions specific to this category can be found in Definitions/Topology.

**Topology** is a geometry of transformations in which the only invariant is continuity.

Some sources suggest that it can indeed be described simply as **the study of continuity**.

As such, it is closely interwoven with the branch of **analysis**.

## Subcategories

This category has the following 90 subcategories, out of 90 total.

### A

### B

### C

### D

### E

### F

### G

### H

### I

### J

### K

### L

### M

### N

### O

### P

### Q

### R

### S

### T

### U

### W

## Pages in category "Topology"

The following 143 pages are in this category, out of 143 total.

### B

- Baire Characterisation Theorem
- Baire-Osgood Theorem
- Banach-Tarski Paradox
- Basis Condition for Coarser Topology
- Basis Condition for Coarser Topology/Corollary 1
- Basis Condition for Coarser Topology/Corollary 2
- Basis has Subset Basis of Cardinality equal to Weight of Space
- Basis induces Local Basis
- Bottom in Ordered Set of Topology
- Boundary of Polygon is Jordan Curve
- Boundary of Polygon is Topological Boundary
- Brouwer's Fixed Point Theorem/One-Dimensional Version
- Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof Using Connectedness

### C

- Cardinality of Image of Mapping of Intersections is not greater than Weight of Space
- Cesàro Mean
- Characterization of Analytic Basis by Local Bases
- Characterization of Interior of Triangle
- Characterization of Prime Element in Inclusion Ordered Set of Topology
- Classification of Compact One-Manifolds
- Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry
- Closed Set in Closed Subspace
- Closed Topologist's Sine Curve is Connected
- Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
- Closure of Subset in Subspace
- Coarseness Relation on Topologies is Partial Ordering
- Complement of Element is Irreducible implies Element is Meet Irreducible
- Composite of Continuous Mappings is Continuous
- Composite of Continuous Mappings is Continuous/Point
- Continuity from Union of Restrictions
- Continuous Mapping is Continuous on Induced Topological Spaces
- Continuous Mapping of Separation
- Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous
- Convex Set is Contractible
- Correspondence between Neighborhood Space and Topological Space

### E

- Element is Meet Irreducible iff Complement of Element is Irreducible
- Empty Set is Element of Topology
- Empty Set Satisfies Topology Axioms
- Equidecomposability is Equivalence Relation
- Equidecomposability Unaffected by Union
- Equidecomposable Nested Sets
- Equivalence of Definitions of Continuous Mapping between Topological Spaces at Point
- Equivalence of Definitions of Continuous Mapping between Topological Spaces/Everywhere
- Equivalence of Definitions of Everywhere Continuous Mapping between Topological Spaces
- Equivalence of Definitions of Finer Topology
- Equivalence of Definitions of Kuratowski Closure Operator
- Equivalence of Definitions of Local Basis
- Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets
- Equivalence of Definitions of Local Basis/Neighborhood Basis of Open Sets Implies Local Basis for Open Sets
- Equivalence of Definitions of Topology
- Equivalence of Definitions of Weight of Topological Space
- Every Filter has Limit Point implies Every Ultrafilter Converges
- Existence and Uniqueness of Generated Topology
- Existence of Local Coordinates
- Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal
- Extendability Theorem for Intersection Numbers
- Extendability Theorem for Intersection Numbers/Corollary

### F

- Filter on Product of Hausdorff Spaces Converges iff Projections Converge
- Filter on Product Space Converges iff Projections Converge
- Filter on Product Space Converges to Point iff Projections Converge to Projections of Point
- Final Topology with respect to Mapping
- Finite Weight Space has Basis equal to Image of Mapping of Intersections
- First Subsequence Rule

### G

### I

### J

### N

### O

### P

- Poincaré Conjecture
- Point Finite Set of Open Sets in Separable Space is Countable
- Positive-Term Generalized Sum Converges iff Supremum
- Preimage Theorem
- Product of Injective Spaces is Injective
- Projection from Product Topology is Continuous
- Projection from Product Topology is Continuous/General Result
- Projection from Product Topology is Open
- Projection from Product Topology is Open/General Result
- Pseudometric induces Topology

### R

- Relational Structure with Topology of Subsets with Property (S) is Topological Space
- Relationship between Limit Inferior and Lower Limit
- Restriction of Continuous Mapping is Continuous/Topological Spaces
- Retract of Injective Space is Injective
- Rokhlin's Theorem (Intersection Forms)
- Rokhlin's Theorem on Bounded Manifolds and Induced Spin Structures

### S

### T

- The sequence lemma
- Thurston's Geometrization Conjecture
- Top in Ordered Set of Topology
- Topological Product of Compact Spaces
- Topological Product of Compact Spaces/Finite Product
- Topological Space induced by Neighborhood Space induced by Topological Space
- Topological Space is Open Neighborhood of Subset
- Topologies Not Always Comparable by Coarseness
- Topologies on Set form Complete Lattice
- Topology as Magma of Sets
- Topology Defined by Basis
- Topology Defined by Closed Sets
- Topology Defined by Neighborhood System
- Topology Discrete iff All Singletons Open
- Topology forms Complete Lattice
- Topology Generated by Closed Sets
- Topology is Locally Compact iff Ordered Set of Topology is Continuous
- Tychonoff's Theorem for Hausdorff Spaces
- Tychonoff's Theorem/General Case