Questions tagged [compactification]

Use this tag for questions about making a topological space into a compact space.

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When a space is NOT locally compact, does it have dense remainder in its compactification?

Any space has to be dense in any of its compactifications (that is part of the definition). Question 1) When the space is NOT locally compact, does that mean that also its remainder in any of its ...
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Are $\mathbb{R}$, $\mathbb{R^n}$ and $l^2$ open, or closed in their compactifications? What about the remainders?

I hope I formulated the question correctly, if not, I will try to update. Thank you for your help. Are $\mathbb{R}$, $\mathbb{R^n}$ and $l^2$ open, or closed in their compactifications? What about the ...
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conformal compactification $\overline G$

Construct a conformal compactification, $\overline G$ of $G:=\Bbb R^{1,1}_*$ and/or provide a diagram of the conformal compactification of $G?$ conformal compactification Let $G$ have the metric ...
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Proving Tychonoff's theorem using the Stone-Cech compactification

Let $X_i, \ i \in I$ be a collection of compact spaces. Consider the product space $X= \prod_i X_i$. Let $βX$ be the Stone-Cech compactification of $X$. For $i \in I$, let $π_i \colon X \to X_i$ be ...
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What is a z-filter?

I am trying to understand meaning of z-filters (def. below). Why are they introduced? Where are they used in mathematics? The only example I have found is that they are used for the construction of ...
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What is the Bohr compactification of a topological group and in particular of the real line? How can you define it?

I have encountered the Bohr compactification in the context of Loop Quantum Cosmology and Polymer Quantum Mechanics and I cannot succeed understanding it in a straightforward way. I just got my Master'...
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1-point compactification of long line (long circle)?

The long line is locally compact and Hausdorff so it has a 1-point compactification. What can be said about it? Is it anything new? Searching internet for "long circle" returns no results, ...
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