# Category:Product Spaces

Jump to navigation
Jump to search

This category contains results about Product Spaces in the context of Topology.

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.

The **product topology** $\tau$ for $S_1 \times S_2$ is the topology with basis $\mathcal B = \left\{{U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}\right\}$.

## Subcategories

This category has only the following subcategory.

## Pages in category "Product Spaces"

The following 53 pages are in this category, out of 53 total.

### C

- Cantor Space as Countably Infinite Product
- Continuous Mapping to Topological Product
- Continuous Mapping to Topological Product/Corollary
- Countable Product of First-Countable Spaces is First-Countable
- Countable Product of Second-Countable Spaces is Second-Countable
- Countable Product of Separable Spaces is Separable
- Countable Product of Sequentially Compact Spaces is Sequentially Compact

### F

- Factor Spaces are T4 if Product Space is T4
- Factor Spaces are T5 if Product Space is T5
- Finite Product of Sigma-Compact Spaces is Sigma-Compact
- Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact
- Finite Product Space is Connected iff Factors are Connected
- Finite Product Space is Connected iff Factors are Connected/Basis for the Induction
- Finite Product Space is Connected iff Factors are Connected/General Case
- Function to Product Space is Continuous iff Composition with Projections are Continuous

### I

### P

- Paracompactness is not always Preserved under Open Continuous Mapping
- Paracompactness is Preserved under Projections
- Points in Product Spaces are Near Open Sets
- Product of Closed Sets is Closed
- Product of Countably Compact Spaces is not always Countably Compact
- Product of Hausdorff Factor Spaces is Hausdorff
- Product of Hausdorff Factor Spaces is Hausdorff/General Result
- Product of Lindelöf Spaces is not always Lindelöf
- Product of Metacompact Spaces is not always Metacompact
- Product of Paracompact Spaces is not always Paracompact
- Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff
- Product Space is Path-connected iff Factor Spaces are Path-connected
- Product Space is T0 iff Factor Spaces are T0
- Product Space is T0 iff Factor Spaces are T0/General Result
- Product Space is T1 iff Factor Spaces are T1
- Product Space is T2 iff Factor Spaces are T2
- Product Space is T3 1/2 iff Factor Spaces are T3 1/2
- Product Space is T3 iff Factor Spaces are T3
- Product Topology is Topology
- Products of Open Sets form Local Basis in Product Space
- Products of Products are Homeomorphic to Collapsed Products
- Projection from Product Topology is Continuous
- Projection from Product Topology is Continuous/General Result

### S

### T

### U

- Uncountable Product of First-Countable Spaces is not always First-Countable
- Uncountable Product of Second-Countable Spaces is not always Second-Countable
- Uncountable Product of Separable Spaces is not always Separable
- Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact
- Universal Property of Product of Topological Spaces