Definition
Continuous Map and Homeomorphism
Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces (Definition~Topological Space).
A map $f : X \to Y$ is continuous if $f^{-1}(U) \in \mathcal{T}_X$ for every $U \in \mathcal{T}_Y$.
If $f$ is a continuous bijection with continuous inverse, it is a homeomorphism, and we write $X \cong Y$.
Depends on
Used in
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
nc9093e38["Topological Space"]
na9fa06fb["Continuous Map and Homeomorphism"]:::current
ndc4990c2["Connectedness and Path-Connectedness"]
n5751451d["Topological Manifold"]
n03a51003["Smooth Map and Diffeomorphism"]
ne5d64c7b["Immersion and Embedding"]
nc5e5fa8f["Covering Space"]
nc9093e38 --> na9fa06fb
na9fa06fb --> ndc4990c2
na9fa06fb --> n5751451d
na9fa06fb --> n03a51003
na9fa06fb --> ne5d64c7b
na9fa06fb --> nc5e5fa8f
click nc9093e38 "../objects/c9093e38.html" "_self"
click ndc4990c2 "../objects/dc4990c2.html" "_self"
click n5751451d "../objects/5751451d.html" "_self"
click n03a51003 "../objects/03a51003.html" "_self"
click ne5d64c7b "../objects/e5d64c7b.html" "_self"
click nc5e5fa8f "../objects/c5e5fa8f.html" "_self"