Category:Definitions/Examples of Topologies
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This category contains definitions of examples of topologies.
Let $S$ be a set.
A topology on $S$ is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the open set axioms:
| \((\text O 1)\) | $:$ | The union of an arbitrary subset of $\tau$ is an element of $\tau$. | |||||||
| \((\text O 2)\) | $:$ | The intersection of any two elements of $\tau$ is an element of $\tau$. | |||||||
| \((\text O 3)\) | $:$ | $S$ is an element of $\tau$. |
If $\tau$ is a topology on $S$, then $\struct {S, \tau}$ is called a topological space.
The elements of $\tau$ are called the open sets of $\struct {S, \tau}$.
Subcategories
This category has the following 37 subcategories, out of 37 total.
A
B
D
- Definitions/Discrete Topology (14 P)
E
F
- Definitions/Final Topology (3 P)
- Definitions/Fort Spaces (5 P)
G
H
I
L
N
- Definitions/Niemytzki Plane (3 P)
O
- Definitions/Ordinal Spaces (13 P)
P
- Definitions/Product Topology (17 P)
Q
R
S
- Definitions/Sorgenfrey Line (1 P)
- Definitions/Stone Spaces (2 P)
T
- Definitions/Thomas's Plank (3 P)
- Definitions/Tychonoff Plank (3 P)
Z
- Definitions/Zariski Topology (7 P)
Pages in category "Definitions/Examples of Topologies"
The following 118 pages are in this category, out of 118 total.
A
C
D
- Definition:Deleted Diameter Topology
- Definition:Deleted Integer Topology
- Definition:Deleted Radius Topology
- Definition:Dieudonné Plank
- Definition:Discrete Extension of Reals
- Definition:Discrete Topology
- Definition:Divisor Topology
- Definition:Double Origin Topology
- Definition:Double Pointed Real Number Line
- Definition:Double Pointed Topology
E
- Definition:Either-Or Space
- Definition:Either-Or Topology
- Definition:Empty Topological Space
- Definition:Euclidean Space
- Definition:Euclidean Space/Euclidean Topology/Real Number Line
- Definition:Evenly Spaced Integer Topology
- Definition:Excluded Point Topology
- Definition:Excluded Set Topology
- Definition:Extended Long Line
- Definition:Extended Real Number Space
F
H
I
- Definition:Identification Topology
- Definition:Included Set Topology
- Definition:Indiscrete Extension of Reals
- Definition:Indiscrete Topology
- Definition:Initial Topology
- Definition:Integer Reciprocal Space
- Definition:Interlocking Interval Topology
- Definition:Irrational Number Space
- Definition:Irrational Slope Topology
- Definition:Irregular Lattice Topology
M
O
P
- Definition:P-adic Topology
- Definition:Parallel Line Topology
- Definition:Parallel Line Topology/Strong
- Definition:Parallel Line Topology/Weak
- Definition:Particular Point Topology
- Definition:Partition Topology
- Definition:Pointed Extension of Reals
- Definition:Pointed Extension of Reals/Irrational
- Definition:Pointed Extension of Reals/Rational
- Definition:Prime Ideal Topology
- Definition:Prime Integer Topology
- Definition:Product Topology
R
S
T
- Definition:Telophase Topology
- Definition:Thomas's Corkscrew
- Definition:Thomas's Plank
- Definition:Topologist's Sine Curve
- Definition:Topology on Extended Complex Plane
- Definition:Topology on Extended Real Numbers
- Definition:Trivial Topological Space
- Definition:Tychonoff Corkscrew
- Definition:Tychonoff Corkscrew/Deleted
- Definition:Tychonoff Plank
- Definition:Tychonoff Plank/Deleted
- Definition:Type Space