277: Net to Product Topological Space Converges to Point iff Each Projection After Net Converges to Component of Point

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A description/proof of that net to product topological space converges to point iff each projection after net converges to component of point

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description 1
• 2: Proof 1
• 3: Description 2
• 4: Proof 2
• 5: Note

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Starting Context

• The reader knows a definition of product topology.
• The reader knows a definition of net with directed index set.
• The reader knows a definition of convergence of net with directed index set.
• The reader admits the proposition that any possibly-infinite-wise product topological space for which the indices set is finite is homeomorphic to the corresponding finite product topological space.

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Target Context

• The reader will have a description and a proof of the proposition that any net to any product topological space converges to a point if and only if the projection to each constituent space after the net converges to the corresponding component of the point.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description 1

For any product topological space, $T={\times }_{\alpha }{T}_{\alpha }$ where $\alpha \in A$ is any possibly uncountable indices set, any net, $n:D\to T$ where $D$ is any directed set, converges to a point, $p=(...,p_{\alpha },...)$ (which is really a function, $f$, such that $f(\alpha )=p_{\alpha }$, which is $limn=p$, if and only if each projection, ${\pi }_{\alpha }:T\to T_{\alpha }$, after $n$ converges to the corresponding component of $p$, $p_{\alpha }$, which is $lim{\pi }_{\alpha }\circ n=p_{\alpha }$.

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2: Proof 1

Let us suppose that $n$ converges to $p$. For any neighborhood, $N_p\subseteq T$, of $p$, there is a $d\in D$ such that for each $d^{\prime }\in D\text{ such that }d\le d^{\prime }$, $n(d^{\prime })\in N_p$. For any neighborhood, $N_{p_{\alpha }}\subseteq T_{\alpha }$, of $p_{\alpha }$, there is an open set, $U_{p_{\alpha }}\subseteq N_{p_{\alpha }}$, around $p_{\alpha }$. For any fixed $\alpha$, let us take the neighborhood, $N_p\subseteq T$, of $p$, that is $N_p={\times }_{\beta \in A}{U}_{\beta }$ where $U_{\beta }=U_{p_{\alpha }}$ when $\beta =\alpha$ and $U_{\beta }=T_{\beta }$ when $\beta \ne \alpha$, which is indeed an open neighborhood of $p$ by the definition of product topology. Then, $n(d^{\prime })\in N_p$, which implies that ${\pi }_{\alpha }\circ n(d^{\prime })\in U_{p_{\alpha }}$. So, ${\pi }_{\alpha }\circ n$ converges to $p_{\alpha }$.

Let us suppose that ${\pi }_{\alpha }\circ n$ converges to $p_{\alpha }$ for each $\alpha \in A$. For each neighborhood, $N_{p_{\alpha }}\subseteq T_{\alpha }$, of $p_{\alpha }$, there is a $d_{\alpha }\in D$ such that for each $d^{\prime }\in D\text{ such that }{d}_{\alpha }\le d^{\prime }$, $n(d^{\prime })\in N_{p_{\alpha }}$. For any neighborhood, $N_p\subseteq T$, of $p$, there is an open set, $U_p\subseteq N_p$, around $p$, where $U_p={\cup }_{\beta }{\times }_{\alpha }{U}_{\beta -\alpha }$ where $\alpha \in A$ where $\beta$ is a member of any possibly uncountable indices set, and for any fixed $\beta$, only finite $U_{\beta -\alpha }$s are not $T_{\alpha }$, by the definition of product topology. There is a $d_{\beta -\alpha }\in D$ such that for each $d^{\prime }\in D\text{ such that }{d}_{\beta -\alpha }\le d^{\prime }$, ${\pi }_{\alpha }\circ n(d^{\prime })\in U_{\beta -\alpha }$. As $D$ is directed, for any fixed $\beta$, there is a $d\in D$ such that $d_{\beta -\alpha }\le d$ for the finite number of $U_{\beta -\alpha }$s, then for each $d^{\prime }\in D\text{ such that }d\le d^{\prime }$, ${\pi }_{\alpha }\circ n(d^{\prime })\in U_{\beta -\alpha }$ for any $\alpha$ for the fixed $\beta$. So, $n(d^{\prime })\in {\times }_{\alpha }{U}_{\beta -\alpha }\subseteq U_p$. So, $n$ converges to $p$.

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3: Description 2

For any product topological space, $T=T_1\times T_2\times ...\times T_n$, any net, $n:D\to T$ where $D$ is any directed set, converges to a point, $p=(p_1,p_2,...,p_n)$, which is $limn=p$, if and only if each projection, ${\pi }_i:T\to T_i$, after $n$ converges to the corresponding component of $p$, $p_i$, which is $lim{\pi }_i\circ n=p_i$.

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4: Proof 2

By the proposition that any possibly-infinite-wise product topological space for which the indices set is finite is homeomorphic to the corresponding finite product topological space, $T$ is homeomorphic to $T^{\prime }={\times }_i{T}_i$ where $i\in \{1,2,...,n\}$, and Description 1 applies to $T^{\prime }$.

If $n$ converges to $p$, the corresponding net, $n^{\prime }:D\to T^{\prime }$, converges to the corresponding point, $p^{\prime }$, because for any neighborhood, $N_{p^{\prime }}\subseteq T^{\prime }$, of $p^{\prime }$, $n$ is eventually in the corresponding neighborhood, $N_p$, of $p$, which implies that $n^{\prime }$ is eventually in $N_{p^{\prime }}$. By Description 1, ${\pi }_i^{\prime }\circ n^{\prime }$ converges to $p_i$ ($p_i^{\prime }$ and $p_i$ are the same). Then, ${\pi }_i\circ n$ converges to $p_i$, because ${\pi }_i^{\prime }\circ n^{\prime }$ and ${\pi }_i\circ n$ are the same.

If ${\pi }_i\circ n$ converges to $p_i$ for each $i$, ${\pi }_i^{\prime }\circ n^{\prime }$ converges to $p_i$. By Description 1, $n^{\prime }$ converges to $p^{\prime }$. Then $n$ converges to $p$, because for any neighborhood, $N_p\subseteq T$, of $p$, $n^{\prime }$ is eventually in the corresponding neighborhood, $N_{p^{\prime }}$, of $p^{\prime }$, which implies that $n$ is eventually in $N_p$.

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5: Note

As is stated in the proposition that any possibly-infinite-wise product topological space for which the indices set is finite is homeomorphic to the corresponding finite product topological space, $T=T_1\times T_2\times ...\times T_n$ and $T^{\prime }={\times }_i{T}_i$ are not exactly the same, as any element of $T$ is like a $⟨p_1,p_2,...,p_n⟩$ while any element of $T^{\prime }$ is a function, $f:\{1,2,...,n\}\to {\cup }_i{T}_i$, although some people may sloppily say that they are the same. While Description 2 seems obvious from the homeomorphism anyway, we have bothered to go more explicit.

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References

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