Category:Examples of Topologies
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This category contains examples of Topology.
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 102 subcategories, out of 102 total.
A
- Alexandroff Plank (empty)
- Alexandroff Square (empty)
- Altered Long Line (empty)
- Appert Space (empty)
B
- Box Topology (8 P)
C
- Complete Order Topologies (1 P)
- Concentric Circle Topology (empty)
D
- Deleted Diameter Topology (empty)
- Deleted Radius Topology (empty)
- Deleted Tychonoff Corkscrew (empty)
- Deleted Tychonoff Plank (empty)
- Dieudonné Plank (empty)
- Divisor Topology (empty)
- Double Origin Topology (empty)
- Double Pointed Real Number Line (empty)
E
- Empty Topological Space (2 P)
- Evenly Spaced Integer Topology (empty)
- Excluded Set Topology (2 P)
- Extended Real Number Space (2 P)
F
- Final Topology (5 P)
- Finite Complement Topology (27 P)
- Fortissimo Space (22 P)
- Furstenberg Topology (3 P)
G
H
- Half-Disc Topology (empty)
- Hewitt's Condensed Corkscrew (empty)
- Hjalmar Ekdal Topology (empty)
I
- Included Set Topology (4 P)
- Integer Reciprocal Space (13 P)
- Interlocking Interval Topology (empty)
- Irrational Number Space (22 P)
- Irrational Slope Topology (empty)
- Irregular Lattice Topology (empty)
L
M
- Maximal Compact Topology (empty)
- Metrizable Tangent Disc Topology (empty)
- Michael's Product Topology (empty)
- Minimal Hausdorff Topology (empty)
- Modified Fort Space (15 P)
- Multiple Pointed Topology (1 P)
N
- Nested Interval Topology (empty)
- Niemytzki Plane (1 P)
O
- Ordinal Spaces (28 P)
- Overlapping Interval Topology (empty)
P
- P-adic Topology (empty)
- Prime Ideal Topology (empty)
Q
R
- Rational Extension in Plane (empty)
- Rational Sequence Topology (empty)
- Relatively Prime Integer Topology (empty)
- Right Order Topology (5 P)
S
- Scott Topology (1 P)
- Simplified Arens Square (empty)
- Smirnov's Deleted Sequence Topology (empty)
- Sorgenfrey Line (12 P)
- Stone Spaces (4 P)
- Strong Operator Topology (empty)
- Strong Parallel Line Topology (empty)
T
- Telophase Topology (empty)
- Thomas's Corkscrew (empty)
- Thomas's Plank (empty)
- Trivial Topological Space (3 P)
W
- Weak Parallel Line Topology (empty)
Z
- Zariski Topology (1 P)
- Z^Z Space (empty)
Pages in category "Examples of Topologies"
The following 2 pages are in this category, out of 2 total.