Category:Definitions/Homeomorphisms
Jump to navigation
Jump to search
This category contains definitions related to Homeomorphisms.
Related results can be found in Category:Homeomorphisms.
Topological Spaces
Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.
Let $f: T_\alpha \to T_\beta$ be a bijection.
$f$ is a homeomorphism if and only if both $f$ and $f^{-1}$ are continuous.
Metric Spaces
The same definition applies to metric spaces:
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a bijection such that:
- $f$ is continuous from $M_1$ to $M_2$
- $f^{-1}$ is continuous from $M_2$ to $M_1$.
Then:
- $f$ is a homeomorphism
- $M_1$ and $M_2$ are homeomorphic.
Subcategories
This category has the following 3 subcategories, out of 3 total.
Pages in category "Definitions/Homeomorphisms"
The following 6 pages are in this category, out of 6 total.