# Category:Topological Subspaces

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This category contains results about **Topological Subspaces**.

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

Let $H \subseteq S$ be a non-empty subset of $S$.

Define:

- $\tau_H := \set {U \cap H: U \in \tau} \subseteq \powerset H$

where $\powerset H$ denotes the power set of $H$.

Then the topological space $T_H = \struct {H, \tau_H}$ is called a **(topological) subspace** of $T$.

## Subcategories

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

## Pages in category "Topological Subspaces"

The following 45 pages are in this category, out of 45 total.

### C

- Closed Set in Topological Subspace
- Closed Set in Topological Subspace/Corollary
- Closure of Subset in Subspace
- Closure of Subset in Subspace/Corollary 1
- Closure of Subset in Subspace/Corollary 2
- Compact in Subspace is Compact in Topological Space
- Compact Subspace of Hausdorff Space is Closed
- Completely Hausdorff Property is Hereditary
- Completely Normal iff Every Subspace is Normal
- Continuity of Composite with Inclusion

### M

### O

### S

- Second-Countability is Hereditary
- Separation Properties Preserved in Subspace
- Separation Properties Preserved in Subspace/Corollary
- Separation Properties Preserved in Subspace/T4 Space
- Sub-Basis for Topological Subspace
- Subspace of Hausdorff Space is Hausdorff
- Subspace of Metrizable Space is Metrizable Space
- Subspace of Product Space has Initial Topology with respect to Restricted Projections
- Subspace of Product Space is Homeomorphic to Factor Space
- Subspace of Subspace is Subspace
- Subspace Topology is Initial Topology with respect to Inclusion Mapping
- Subspace Topology on Initial Topology is Initial Topology on Restrictions

### T

- T0 Property is Hereditary
- T1 Property is Hereditary
- T2 Property is Hereditary
- T3 1/2 Property is Hereditary
- T3 Property is Hereditary
- T4 Property is not Hereditary
- T4 Property Preserved in Closed Subspace
- T4 Property Preserved in Closed Subspace/Corollary
- T5 Property is Hereditary
- T5 Space iff Every Subspace is T4
- Topological Subspace is Topological Space