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Each will have an introduction, which will tell us a little about their history and what makes this park special. They will also include an overview of the attractions on offer and a photo gallery. The first park we will visit is Millbrook Park and Secret Gardens. This park is on the border of Bedfordshire and Hertfordshire. Millbrook Park is open daily year-round from 9.00am to sunset. It is free to enter and is a great walk through the park which also has a mixture of spaces designed for walking, running, football or simply enjoying the surrounding countryside. It has also been featured in the Heritage Lottery Fund’s National Register of Historic Parks and Gardens and was made a National Trust site in 2016.
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Practical example for the Tietze Extension Theorem
I’ve been trying to wrap my head around the proofs I’ve seen for the following theorem, and was hoping someone could point me to a practical example. I’ll restate it here:
Theorem: Let $X$ be a locally compact Hausdorff space and let $Y \subseteq X$ be open. Let $K$ be an arbitrary compact set of $X$. Then there exists an open set $U \subseteq Y$ such that $K \subseteq U$ and $U \cap K
eq \emptyset$.
Proof: Let $K$ be compact and let $K^c$ be its complement. Then $K^c$ is compact as well, so we can find compact $C \subseteq K^c$ such that $C \cap K^c = \emptyset$. Then $K \cap C = \emptyset$, which is what we want.
So far so good. But I really want to be able to understand this. Why does this work? This feels really abstract to me. In a statement like: “Let $X$ be a locally compact Hausdorff space and let $Y \subseteq X$ be open” I feel like I know what I’m supposed to expect, but for this it’s just mind-blowing. How does it know that $K^c$ is compact and so can find $C$? Is it a general property of locally compact Hausdorff spaces? Why does it work?
I’d greatly appreciate it if people could please take the time to thoroughly explain how something like this actually follows from the axioms for locally compact Hausdorff spaces. I’m having a really hard time seeing the details.
A:
It isn’t a property of locally compact Hausdorff spaces. The problem comes from the fact that your statement refers to two things:
$X$ is a locally compact Hausdorff space
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