Abhijeet Mulgund's Personal Webpage

Open

Last updated Nov 1, 2022

# Definition

Suppose $(X, \tau)$ is a Topological Space. Then $U \subset X$ is Open if $U \in \tau$.

Backlinks

A Compact Set that is not Closed
A Connected Set is Pathwise Connected in a Metric Space
A Function is Continuous iff it preserves Net Convergence
A Function is Continuous iff it takes Closed Sets back to Closed Sets
A Limit Point that is not a Sequence Limit
A Set is Open in the Metric Topology iff it contains a Ball around each Point
A Space with Connected Components that are not Open
A Topological Basis can be thinned out by a smaller one
A Union of all Neighborhood Bases is a Topological Basis
A point in a First Countable Space is a Limit Point iff it is a Sequential Limit
A point is a Limit Point iff it is a Net Limit
All points in a Set are Limit Points
Closed
Closed Ball is Closed
Closed Intervals are Closed
Closed Subset of a Compact Set is Compact
Closure of a Set is all its Limit Points
Cocountable Topology
Compact Sets are Separable from Points in a Hausdorff Space
Compact Sets in Hausdorff Spaces are Closed
Compact Spaces are Complete
Composition of Continuous Functions is Continuous
Connected
Connected Components of Locally Path-Connected Spaces are Open
Connected Components of Topological Manifolds are Open
Continuous Functions Preserve Compactness
Continuous Functions Preserve Connectedness
Continuous Functions Preserve Limit Points
Continuous Functions Preserve Sequence Limits
Continuous Open Maps transfer Bases
Coordinate Ball
Coordinate Balls form a Basis for Manifolds
Discrete Metric induces the Discrete Topology
Discrete Topology
Disjoint Compact Sets are Separable from each other in a Hausdorff Space
Equivalent Conditions for Local Compactness of Hausdorff Spaces
Equivalent Definitions of Locally Euclidean
Euclidean Space is Second Countable
Every Connected Component is Closed
Every Open Subset of Locally Path-Connected Space is Locally Path-Connected
Graphs of Continuous Functions are Topological Manifolds
Hausdorff
House Rules and Conventions
If Connected Sets share a point, then their Union is Connected
Intersection of Closed Sets is Closed
Limit Point
Limit Points of a subset are Limit Points of the original Set
Locally Compact
Neighborhood Basis
Neighborhood of a Point
Net Convergence
Net Limits are unique in Hausdorff Spaces
Open
Open Cover
Open Halfspace
Open Map
Monotone Nets Converge to their Supremum
Path-Connected Components are equal to the Connected Components in a Locally Path-Connected Space
Path-Connected Components of Locally Path-Connected Spaces are Open
Path-Connected Components of Topological Manifolds are Open
Product Topological Space
Product of Topological Manifolds is a Manifold
Sequence Convergence
Sequence Limits are unique in Hausdorff Spaces
Sequential Limits are Limit Points of the Sequence
Slices of a Continuous Function are Continuous
Subspace Topology
Support of a Measure
Support of a Measure is Closed
Supremum is in the Closure of a Nonempty Set
Topological Basis
Topological Embedding

Open

# Definition

Backlinks

Interactive Graph