# Category:Product Spaces

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

Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$:

- $\ds S := \prod_{i \mathop \in I} S_i$

Let $\tau$ be the product topology on $S$.

The topological space $\struct {X, \tau}$ is called the **product space** of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Product Spaces"

The following 39 pages are in this category, out of 39 total.

### C

### 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

### P

- Paracompactness is not always Preserved under Open Continuous Mapping
- Paracompactness is Preserved under Projections
- Product of Closed and Half-Open Unit Intervals is Homeomorphic to Product of Half-Open Unit Intervals
- 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 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 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 1/2 iff Factor Spaces are T3 1/2/Factor Spaces are T3 1/2 implies Product Space is T3 1/2
- Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Product Space is T3 1/2 implies Factor Spaces are T3 1/2
- Product Space is T3 iff Factor Spaces are T3
- Product Space Local Basis Induced from Factor Spaces Local Bases
- Products of Products are Homeomorphic to Collapsed Products