# 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 95 subcategories, out of 95 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 112 pages are in this category, out of 112 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

### 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
- Closed Set in Closed Subspace
- Closed Topologist's Sine Curve is Connected
- Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
- Coarseness Relation on Topologies is Partial Ordering
- Complement of Element is Irreducible implies Element is Meet Irreducible
- Composite of Continuous Mappings is Continuous
- 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 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/Corollary

### F

### G

### I

### J

### N

### O

### P

### R

### S

### T

- 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 are not necessarily Comparable by Coarseness
- Topologies on Set form Complete Lattice
- Topology as Magma of Sets
- 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/General Case