# Category:Product Spaces

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

Let $\struct{S_1, \tau_1}$ and $\struct{S_2, \tau_2}$ be topological spaces.

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

Let $\tau$ be the Tychonoff topology on $S_1 \times S_2$.

From Natural Basis of Tychonoff Topology of Finite Product, $\tau$ is the topology generated by the basis

- $\BB = \set{U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}$.

The topological space $\struct{S_1 \times S_2, \tau}$ is called the **product space** of $\struct{S_1, \tau_1}$ and $\struct{S_2, \tau_2}$.

## Subcategories

This category has only the following subcategory.

## Pages in category "Product Spaces"

The following 54 pages are in this category, out of 54 total.

### C

- Cantor Space as Countably Infinite Product
- Continuous Mapping to Product Space
- Continuous Mapping to Topological Product
- Continuous Mapping to Topological Product/Corollary
- Continuous Mapping to Topological Product/General Result
- 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

### 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 Product in Category of Topological Spaces
- 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
- 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