# 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 142 subcategories, out of 142 total.

### A

- Accumulation Points (8 P)

### B

- Baire Property (empty)
- Banach-Tarski Paradox (9 P)

### C

- Category of Open Sets (empty)
- Cells (Topology) (empty)
- Clopen Sets (15 P)
- Cobordisms (empty)
- Cohomology Groups (empty)
- Compactifications (empty)
- Condensation Points (10 P)
- Convergent Sequences (Topology) (empty)

### D

- Derived Sets (empty)
- Differential Topology (2 P)
- Dimension Theory (empty)
- Discrete Sets of Subsets (2 P)

### E

- Embeddings (Topology) (empty)
- Everywhere Dense (1 P)

### F

- F-Sigma Sets (13 P)

### G

- G-Delta Sets (17 P)
- Geometric Topology (empty)

### H

- Hausdorff Paradox (6 P)

### I

### J

- Jordan Arcs (2 P)
- Jordan Curves (12 P)

### K

- Klein Bottle (empty)

### L

- Loops (Topology) (5 P)

### M

- Morse Theory (empty)

### N

- Neighborhood Bases (5 P)
- Nets (Topology) (empty)
- Nowhere Dense (1 P)

### O

- Omega-Accumulation Points (8 P)
- Oscillation (5 P)

### P

- Paths (Topology) (1 P)
- Perfect Sets (3 P)
- Polish Spaces (empty)
- Proper Mappings (empty)

### Q

### R

- Regular Closed Sets (4 P)
- Regular Open Sets (4 P)

### S

- Schemes (6 P)
- Set Derivatives (13 P)
- Simply Connected Spaces (11 P)
- Sub-Bases (2 P)
- Submanifolds (empty)

### T

- Tangent Bundles (2 P)
- Topological Order Theory (34 P)
- Topological Sum (2 P)

### U

### W

- Weakly Hereditary Properties (1 P)

## 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 Topological Boundary
- Brouwer's Fixed Point Theorem

### 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
- Characterization of Zero Entropy of Open Cover
- 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
- Continuous Image of Path-Connected Set is Path-Connected
- Continuous Mapping is Continuous on Induced Topological Spaces
- 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 Convergent Sequence in Topology
- 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 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

- Sequence Lemma
- Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point
- Set of Invertible Continuous Transformations is Open Subset of Continuous Linear Transformations in Supremum Operator Norm Topology
- Shift of Finite Type is Compact
- Subsets of Equidecomposable Subsets are Equidecomposable

### 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 forms Complete Lattice
- Topology Generated by Closed Sets
- Topology is Locally Compact iff Ordered Set of Topology is Continuous