generated from smyalygames/quartz
13 lines
1.0 KiB
Markdown
13 lines
1.0 KiB
Markdown
# Definition
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Let $X$ be a topological space
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1. A **neighbourhood** of $x \in X$ is an [[Open Sets|open set]] $A$ with $x \in A$.
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2. $X$ is Hausdorff if for $x \neq y\; \exists \, \text{neighbourhoods} \, A, B$ of $x$ and $y$ respectively, such that $A \cap B = \emptyset$.
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3. $A \subset X$ is **closed** if $A^{\complement}$ is open.
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4. The **closure** (denoted by a bar over the set) $\overline{Y}$ of a subset $Y \subset X$ is the intersection of all closed subsets of $X$ that contain $Y$.
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5. $X$ is **[[Compact|compact]]** if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]].
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6. $X$ is **locally compact** if any $x \in X$ has a neighbourhood with compact closure.
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7. $X$ is $\sigma$-compact if it is a countable union of compact subsets with respect to the **relative topology**, i.e. an [[Open Sets|open set]] of a subset $Z$ of $X$ is of the type $Z \cap A$ where $A$ is open in $X$.
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## Dense
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Say $Y$ is **dense** in $X$ if $\overline{Y} = X$. If $Y$ is countable.
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## Separable
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Say $\overline{Y} = X$, then we say that $X$ is **separable**. |