# Category:Homeomorphisms (Topological Spaces)

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This category contains results about **homeomorphisms** in the context of **topological spaces**.

Definitions specific to this category can be found in **Definitions/Homeomorphisms (Topological Spaces)**.

$f$ is a **homeomorphism** if and only if both $f$ and $f^{-1}$ are continuous.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### E

- Embeddings (Topology) (empty)

## Pages in category "Homeomorphisms (Topological Spaces)"

The following 52 pages are in this category, out of 52 total.

### C

- Cartesian Product of Homeomorphisms is Homeomorphism
- Compact Convex Set with Nonempty Interior is Homeomorphic to Cone on Boundary
- Compact Convex Sets with Nonempty Interior are Homeomorphic
- Completely Hausdorff Space is Preserved under Homeomorphism
- Completely Normal Space is Preserved under Homeomorphism
- Composite of Homeomorphisms is Homeomorphism
- Cones on Homeomorphic Spaces are Homeomorphic
- Continuous Bijection from Compact to Hausdorff is Homeomorphism
- Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
- Continuous Involution is Homeomorphism

### H

- Hausdorff Condition is Preserved under Homeomorphism
- Homeomorphic Image of Local Basis is Local Basis
- Homeomorphic Image of Neighborhood Basis is Neighborhood Basis
- Homeomorphic Image of Nowhere Dense Set is Nowhere Dense
- Homeomorphic Image of Sub-Basis is Sub-Basis
- Homeomorphic Non-Comparable Particular Point Topologies
- Homeomorphic Topologies on Same Set may not be Identical
- Homeomorphic Topology of Initial Topology is Initial Topology
- Homeomorphism between Topological Spaces may not be Unique
- Homeomorphism iff Image of Closure equals Closure of Image
- Homeomorphism may Exist between Non-Comparable Topologies
- Homeomorphism Relation is Equivalence

### I

### P

### R

### S

### T

- T0 Space is Preserved under Homeomorphism
- T1 Space is Preserved under Homeomorphism
- T3 1/2 Space is Preserved under Homeomorphism
- T3 Space is Preserved under Homeomorphism
- T4 Space is Preserved under Homeomorphism
- T5 Space is Preserved under Homeomorphism
- Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic
- Topologies on Set with More than One Element may not be Homeomorphic
- Tychonoff Space is Preserved under Homeomorphism