# Category:Open Sets

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This category contains results about **Open Sets** in the context of **Topology**.

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

Then the elements of $\tau$ are called the **open sets of $T$**.

Thus:

**$U \in \tau$**

and:

**$U$ is open in $T$**

are equivalent statements.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

### C

- Clopen Sets (15 P)

### M

### O

- Open Ball is Open Set (4 P)
- Open Sets (Metric Spaces) (1 P)

### R

- Regular Open Sets (4 P)

### S

## Pages in category "Open Sets"

The following 56 pages are in this category, out of 56 total.

### C

- Catesian Product of Open Real Intervals is Open in Real Number Plane
- Characterization of Open Set by Open Cover
- Closed Real Interval is not Open Set
- Closure of Open Real Interval is Closed Real Interval
- Closure of Open Set of Closed Extension Space
- Closure of Open Set of Particular Point Space
- Complement of Closed Set is Open Set
- Complement of Lower Closure of Element is Open in Scott Topological Ordered Set
- Continuous Mapping on Union of Open Sets

### E

### I

- Image of Open Set under Continuous Mapping in Metric Space may not be Open
- Image of Point under Open Neighborhood of Diagonal is Open Neighborhood of Point
- Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset
- Infinite Intersection of Open Sets of Metric Space may not be Open
- Interior of Open Set

### O

- Open and Closed Sets in Multiple Pointed Topology
- Open Ball is Open Set
- Open Balls form Basis for Open Sets of Metric Space
- Open iff Upper and with Property (S) in Scott Topological Lattice
- Open Real Interval is Open Set
- Open Real Interval is Open Set/Corollary
- Open Real Interval is Regular Open
- Open Set Disjoint from Set is Disjoint from Closure
- Open Set is G-Delta Set
- Open Set Less One Point is Open
- Open Set Less One Point is Open/Corollary
- Open Set may not be Open Ball
- Open Set minus Closed Set is Open
- Open Sets in Metric Space
- Open Sets in Real Number Line
- Open Sets of Cartesian Product of Metric Spaces under Chebyshev Distance