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Questions about Tychonoff spaces natural embedding.
$begingroup$
On Wikipedia it states that for any Tychonoff space $X$ there is a natural embedding into $[0,1]^{C(X,[0,1])}$. I assume this embedding is $iota(x)(f)=f(x)$. I am able to prove that $iota$ is continuous and injective. However, I am not able to see why $iota$ is open onto its image.
I have one more question about this embedding, since Wikipedia also states this is the Stone Čech compactification. However I do not see why $iota(X)$ is dense. Wouldn't the Stone Čech compactification be $beta X=overline{iota(X)}$?
compactness product-space
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
$endgroup$
add a comment |
$begingroup$
On Wikipedia it states that for any Tychonoff space $X$ there is a natural embedding into $[0,1]^{C(X,[0,1])}$. I assume this embedding is $iota(x)(f)=f(x)$. I am able to prove that $iota$ is continuous and injective. However, I am not able to see why $iota$ is open onto its image.
I have one more question about this embedding, since Wikipedia also states this is the Stone Čech compactification. However I do not see why $iota(X)$ is dense. Wouldn't the Stone Čech compactification be $beta X=overline{iota(X)}$?
compactness product-space
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
$endgroup$
add a comment |
$begingroup$
On Wikipedia it states that for any Tychonoff space $X$ there is a natural embedding into $[0,1]^{C(X,[0,1])}$. I assume this embedding is $iota(x)(f)=f(x)$. I am able to prove that $iota$ is continuous and injective. However, I am not able to see why $iota$ is open onto its image.
I have one more question about this embedding, since Wikipedia also states this is the Stone Čech compactification. However I do not see why $iota(X)$ is dense. Wouldn't the Stone Čech compactification be $beta X=overline{iota(X)}$?
compactness product-space
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
$endgroup$
On Wikipedia it states that for any Tychonoff space $X$ there is a natural embedding into $[0,1]^{C(X,[0,1])}$. I assume this embedding is $iota(x)(f)=f(x)$. I am able to prove that $iota$ is continuous and injective. However, I am not able to see why $iota$ is open onto its image.
I have one more question about this embedding, since Wikipedia also states this is the Stone Čech compactification. However I do not see why $iota(X)$ is dense. Wouldn't the Stone Čech compactification be $beta X=overline{iota(X)}$?
compactness product-space
compactness product-space
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
edited Jan 20 at 5:34
SmileyCraft
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 4:25
SmileyCraftSmileyCraft
3,476518
3,476518
add a comment |
add a comment |
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