An simple example to show that every countably compact space needn't be compact

Question

I am willing to study compact and connected in topological space and apply in other topological spaces. I am a beginner in this subject. Kindly give some examples. I have went through few books but I couldn't get clear idea.

Answer 1

Let $X$ be a compact space. Take a cardinal $\lambda$ and endow a set $X^\lambda$ by Tychonoff product topology. By Tychonoff Theorem, a space $X^\lambda$ is compact. Choose a point $x’\in X$, and consider a so-called $\Sigma$-product.

$$\Sigma(X,x’)=\{(x_\alpha)_{\alpha<\lambda}\in X^\lambda: |\{\alpha: x_\alpha\ne x’\}|\le\omega\}.$$

Then the space $\Sigma(X,x’)$ is countable compact dense subspace of $X^\lambda$. But if the space $X$ is Hausdorff, contains at least two points, and the cardinal $\lambda$ is uncountable then $\Sigma(X,x’)\ne X^\lambda$ and it is not compact.