Category:Definitions/Denseness
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This category contains definitions related to Denseness in the context of Topology.
Related results can be found in Category:Denseness.
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a subset.
Definition 1
The subset $H$ is (everywhere) dense in $T$ if and only if:
- $H^- = S$
where $H^-$ is the closure of $H$.
Definition 2
The subset $H$ is (everywhere) dense in $T$ if and only if the intersection of $H$ with every non-empty open set of $T$ is non-empty:
- $\forall U \in \tau \setminus \set \O: H \cap U \ne \O$
Definition 3
The subset $H$ is (everywhere) dense in $T$ if and only if every neighborhood of every point of $S$ contains at least one point of $H$.
Subcategories
This category has the following 3 subcategories, out of 3 total.
D
- Definitions/Densely Ordered (6 P)
E
- Definitions/Everywhere Dense (12 P)
N
- Definitions/Nowhere Dense (3 P)
Pages in category "Definitions/Denseness"
The following 4 pages are in this category, out of 4 total.