Compactness Properties Preserved under Projection Mapping

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Theorem

Let $I$ be an indexing set with countable cardinality.

Let $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \left({S, \tau}\right) = \prod_{\alpha \mathop \in I} \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$.


Let $\operatorname{pr}_\alpha: \left({S, \tau}\right) \to \left({S_\alpha, \tau_\alpha}\right)$ be the projection on the $\alpha$ coordinate.

Then $\operatorname{pr}_\alpha$ preserves the following compactness properties.

That is, if $\left({S, \tau}\right)$ has one of the following properties, then each of $\left({S_\alpha, \tau_\alpha}\right)$ has the same property:

Compact Space
$\sigma$-Compact Space
Countably Compact Space
Sequentially Compact Space
Lindelöf Space
Locally Compact Space
Weakly Locally Compact Space
Paracompact Space


Proof

First note that Projection from Product Topology is Continuous.

Also note that Projection is Surjection.

It follows from Compactness Properties Preserved under Continuous Surjection that:

Compact Space
$\sigma$-Compact Space
Countably Compact Space
Sequentially Compact Space
Lindelöf Space

are all preserved under projections.


Next note that Projection from Product Topology is Open.

It follows from Local Compactness is Preserved under Open Continuous Surjection that local compactness is preserved under projections.

It follows from Weak Local Compactness is Preserved under Open Continuous Surjection that weak local compactness is preserved under projections.

The final result is that Paracompactness is Preserved under Projections.



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