# Category:Hausdorff Spaces

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This category contains results about **$T_2$ (Hausdorff) spaces**.

$\struct {S, \tau}$ is a **Hausdorff space** or **$T_2$ space** if and only if:

- $\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$

That is:

- for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.

## Subcategories

This category has the following 14 subcategories, out of 14 total.

## Pages in category "Hausdorff Spaces"

The following 89 pages are in this category, out of 89 total.

### A

### C

- Characterization of Hausdorff Property in terms of Nets
- Closed Extension Topology is not Hausdorff
- Closure Condition for Hausdorff Space
- Compact Complement Space is not T2, T3, T4 or T5
- Compact Hausdorff Space is Locally Compact
- Compact Hausdorff Space is T4
- Compact Hausdorff Space with no Isolated Points is Uncountable/Lemma
- Compact Hausdorff Topology is Maximally Compact
- Compact Hausdorff Topology is Minimal Hausdorff
- Compact Subspace of Hausdorff Space is Closed
- Compactness Properties in Hausdorff Spaces
- Completely Hausdorff Space is Hausdorff Space
- Condition for Alexandroff Extension to be T2 Space
- Continuous Bijection from Compact to Hausdorff is Homeomorphism
- Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
- Continuous Function from Compact Hausdorff Space to Itself Fixes a Non-Empty Set
- Continuous Mapping from Compact Space to Hausdorff Space is Closed Mapping
- Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness
- Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide
- Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 2
- Convergent Sequence in Hausdorff Space has Unique Limit
- Countable Complement Space is not T2

### D

### E

- Equal Images of Mappings to Hausdorff Space form Closed Set
- Equivalence of Definitions of T2 Space
- Existence of Compact Hausdorff Space which is not T5
- Existence of Hausdorff Space which is not Completely Hausdorff
- Existence of Hausdorff Space which is not T3, T4 or T5
- Existence of Maximal Compact Topological Space which is not Hausdorff
- Existence of Minimal Hausdorff Space which is not Compact

### F

- Factor Spaces of Hausdorff Product Space are Hausdorff
- Finite Complement Space is not T2
- Finite Subspace of Hausdorff Space is Closed
- First Subsequence Rule
- First-Countable Space is Hausdorff iff All Convergent Sequences have Unique Limit
- Fixed Point Set of Continuous Self-Map on Hausdorff Space is Closed

### G

### H

### I

- Image of Group Homomorphism is Hausdorff Implies Kernel is Closed
- Initial Topology with respect to Point-Separating Family of Mappings onto Hausdorff Spaces is Hausdorff
- Inner Limit in Hausdorff Space by Open Neighborhoods
- Inner Limit in Hausdorff Space by Set Closures
- Intersection of Open Sets of Hausdorff Space containing Point is Singleton
- Irreducible Hausdorff Space is Singleton

### M

### N

### P

### S

- Sierpiński's Theorem
- Sorgenfrey Line is Hausdorff
- Space in which All Convergent Sequences have Unique Limit not necessarily Hausdorff
- Space such that Intersection of Open Sets containing Point is Singleton may not be Hausdorff
- Subgroup is Closed iff Quotient is Hausdorff
- Subspace of Hausdorff Space is Hausdorff

### T

- T2 Property is Hereditary
- T2 Space is Preserved under Closed Bijection
- T2 Space is T1 Space
- T3 1/2 Space is not necessarily T2 Space
- Topological Group is Hausdorff iff Identity is Closed
- Topological Group is T1 iff T2
- Topological Vector Space is Hausdorff iff T1
- Tychonoff Space is Regular, T2 and T1
- Tychonoff's Theorem for Hausdorff Spaces