# Category:T1 Spaces

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This category contains results about **$T_1$ (Fréchet) spaces** in the context of **Topology**.

$\struct {S, \tau}$ is a **Fréchet space** or **$T_1$ space** if and only if:

- $\forall x, y \in S$ such that $x \ne y$, both:
- $\exists U \in \tau: x \in U, y \notin U$

- and:
- $\exists V \in \tau: y \in V, x \notin V$

## Subcategories

This category has the following 5 subcategories, out of 5 total.

### C

- Completely Normal Spaces (15 P)

### F

- Finite T1 Space is Discrete (3 P)
- Fully Normal Spaces (4 P)

### N

### T

## Pages in category "T1 Spaces"

The following 48 pages are in this category, out of 48 total.

### C

- Characterization of T1 Space using Basis
- Characterization of T1 Space using Neighborhood Basis
- Closed Extension Topology is not T1
- Closure of Dense-in-itself is Dense-in-itself in T1 Space
- Closure of Derivative is Derivative in T1 Space
- Compact Complement Topology is T1
- Condition for Alexandroff Extension to be T1 Space
- Countable Complement Space is T1

### D

### E

### F

### N

### T

- T0 Space is not necessarily T1 Space
- T1 Property is Hereditary
- T1 Space is Preserved under Closed Bijection
- T1 Space is Preserved under Homeomorphism
- T1 Space is T0 Space
- T1 Space is T1/2 Space
- T1 Space is Weakly Countably Compact iff Countably Compact
- T2 Space is T1 Space
- Topological Group is T1 iff T2
- Topological Vector Space is Hausdorff iff T1
- Totally Disconnected Space is T1
- Tychonoff Space is Regular, T2 and T1