Category:Subspace of Product Space is Homeomorphic to Factor Space
This category contains pages concerning Subspace of Product Space is Homeomorphic to Factor Space:
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.
Let $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.
Suppose that $X$ is non-empty.
Then for each $i \in I$ there is a subspace $Y_i \subseteq X$ which is homeomorphic to $\struct {X_i, \tau_i}$.
Specifically, for any $z \in X$, let:
- $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$
and let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.
Then $\struct {Y_i, \upsilon_i}$ is homeomorphic to $\struct {X_i, \tau_i}$, where the homeomorphism is the restriction of the projection $\pr_i$ to $Y_i$.
Pages in category "Subspace of Product Space is Homeomorphic to Factor Space"
The following 14 pages are in this category, out of 14 total.
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- Subspace of Product Space is Homeomorphic to Factor Space
- Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton
- Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Lemma
- Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Proof 1
- Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Proof 2
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 1
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 1
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 2
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 2
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Continuous Mapping
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Injection
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Lemma 1
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Open Mapping
- Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Surjection