Category:Connectedness

A

• [×] Arc-Connectedness‎ (8 P)

B

• [×] Biconnectedness‎ (1 P)

C

• [×] Connectedness Between Two Points‎ (8 P)
• [+] Continua‎ (1 C)

D

• [×] Dispersion Points‎ (2 P)

E

• [×] Extremally Disconnected Spaces‎ (8 P)

H

• [×] Hyperconnectedness‎ (14 P)

L

• [+] Local Connectedness‎ (2 C, 13 P)

P

• [×] Path-Connectedness‎ (15 P)
• [×] Punctiform Spaces‎ (1 P)

S

• [×] Scattered Spaces‎ (12 P)

T

• [×] Totally Disconnected Spaces‎ (8 P)

T cont.

• [×] Totally Pathwise Disconnected Spaces‎ (2 P)
• [×] Totally Separated Spaces‎ (9 P)

U

• [×] Ultraconnectedness‎ (10 P)

Z

• [×] Zero Dimensional Spaces‎ (7 P)

A

• Arens-Fort Space is Not Connected

C

• Clopen Set contains Components of All its Points
• Compact Complement Topology is Connected
• Condition on Connectedness by Clopen Sets
• Connected iff no Proper Clopen Sets
• Connected Open Subset of Euclidean Space is Path-Connected
• Connected Space is Connected Between Two Points
• Connectedness of Points is Equivalence Relation
• Continuous Image of Connected Space is Connected
• Countable Complement Space is Connected

E

• Equivalence of Connectedness Definitions

E cont.

• Equivalence of Definitions of Component
• Excluded Point Space is Connected

F

• Finite Complement Space is Connected

H

• Hyperconnected Space is Connected

I

• Indiscrete Space is Connected
• Intermediate Value Theorem (Topology)

N

• Non-Trivial Connected Set in T1 Space is Dense-in-itself
• Non-Trivial Discrete Space Not Connected

O

• Only Intervals are Connected
• Open Extension Space is Connected

P

• Path-Connected Space is Connected

P cont.

• Product Space is Connected iff Factors are Connected

R

• Relationship between Component Types

S

• Sequence of Implications of Connectedness Properties
• Sequence of Implications of Disconnectedness Properties
• Sets in Modified Fort Space are Disconnected
• Spaces with Connected Intersection have Connected Union
• Subset of Closure of Connected Subspace

T

• Topological Space with One Quasicomponent is Connected
• Totally Disconnected but Connected Set must be Singleton

U

• Union of Connected Sets with Non-Null Intersection is Connected