# The Stone-Čech compactification of a inverse system

Is the Stone-Čech compactification of the inverse limit of an inverse system $$\left\{ X_{i},f_{ij},I\right\}$$ of Tychonoff spaces equal to the limit of the inverse system $$\left\{ \beta X_{i},\beta f_{ij},I\right\}$$, where $$\beta f_{ij}$$ is the extension of $$f_{ij}$$ over $$\beta X_{i}$$ and $$\beta X_{j}$$?

Similarly, is the Hewitt realcompactification of the limit of an inverse system $$\left\{ X_{i},f_{ij},I\right\}$$ of Tychonoff spaces equal to the limit of the inverse system $$\left\{ \upsilon X_{i},\upsilon f_{ij},I\right\}$$, where $$\upsilon X_{i}$$ is the realcompactification of $$X_{i}$$?

• Well, it isn’t even true that compactifications of products are products of compactifications. – user131781 Nov 4 '19 at 10:53
• How does this give a contradiction? – Mehmet Onat Jan 1 at 19:53

Let $$X_n$$ be $$\{k\in\mathbb{N}:k\ge n\}$$ and let $$f_n:X_{n+1}\to X_n$$ be the inclusion map. The inverse limit of the system $$\{X_n,f_n,\mathbb{N}\}$$ is empty; the limit of the system $$\{\beta X_n,\beta f_n,\mathbb{N}\}$$ is $$\beta\mathbb{N}\setminus\mathbb{N}$$. Similarly for $$\alpha\in\omega_1$$ let $$Y_\alpha$$ be $$\omega_1\setminus\alpha$$, with $$g_\alpha:Y_{\alpha+1}\to Y_\alpha$$ the inclusion map. The inverse limit of the $$Y_\alpha$$s is empty; as $$\upsilon Y_\alpha=[\alpha,\omega_1]$$ for all $$\alpha$$, the inverse limit of the $$\upsilon Y_\alpha$$s consists of one point, $$\omega_1$$.