# Category:Connected Spaces

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This category contains results about Connected Spaces in the context of Topology.

Definitions specific to this category can be found in Definitions/Connected Spaces.

Let $T = \struct {S, \tau}$ be a non-empty topological space.

$T$ is **connected** if and only if there exists no continuous surjection from $T$ onto a discrete two-point space.

## Subcategories

This category has the following 23 subcategories, out of 23 total.

### A

### B

- Biconnected Sets (3 P)

### C

- Components (10 P)
- Cut Points (1 P)

### D

### E

### I

### L

### P

- Path-Connected Sets (3 P)

### Q

- Quasicomponents (5 P)

### S

- Sierpiński's Theorem (2 P)

### T

## Pages in category "Connected Spaces"

The following 57 pages are in this category, out of 57 total.

### C

- Compact Complement Topology is Connected
- Components are Open iff Union of Open Connected Sets
- Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets
- Components are Open iff Union of Open Connected Sets/Lemma 1
- Components are Open iff Union of Open Connected Sets/Space is Union of Open Connected Sets implies Components are Open
- Connected and Locally Path-Connected Implies Path Connected
- Connected iff no Proper Clopen Sets
- Connected Open Subset of Euclidean Space is Path-Connected
- Connected Set in Subspace
- Connected Space is Connected Between Two Points
- Connected Space is not necessarily Locally Connected
- Connected Subset of Union of Disjoint Open Sets
- Connected Subspace of Linearly Ordered Space
- Connectedness of Points is Equivalence Relation
- Continuous Image of Connected Space is Connected
- Continuous Image of Connected Space is Connected/Corollary 1
- Continuous Image of Connected Space is Connected/Corollary 2
- Countable Complement Space is Connected
- Countable Union of Overlapping Connected Sets is Connected

### E

- Equivalence of Definitions of Connected Topological Space
- Equivalence of Definitions of Irreducible Space/3 iff 7
- Equivalence of Definitions of Ultraconnected Space/1 iff 3
- Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point
- Excluded Point Space is Connected
- Existence of Connected Non-T1 Scattered Space
- Existence of Connected Punctiform Space
- Existence of Connected Space which is Totally Pathwise Disconnected

### F

### I

### L

### R

### S

### U

- Ultraconnected Space is Connected
- Union of Connected Sets with Common Point is Connected
- Union of Connected Sets with Non-Empty Intersections is Connected
- Union of Connected Sets with Non-Empty Intersections is Connected/Corollary
- Union of Path-Connected Sets with Common Point is Path-Connected
- Unit Interval is Path-Connected in Real Numbers