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authorJustin Gassner <justin.gassner@mailbox.org>2024-02-15 05:11:07 +0100
committerJustin Gassner <justin.gassner@mailbox.org>2024-02-15 05:11:07 +0100
commit7c66b227a494748e2a546fb85317accd00aebe53 (patch)
tree9c649667d2d024b90b32d36ca327ac4b2e7caeb2 /pages/general-topology/topological-spaces.md
parent28407333ffceca9b99fae721c30e8ae146a863da (diff)
downloadsite-7c66b227a494748e2a546fb85317accd00aebe53.tar.zst
Update
Diffstat (limited to 'pages/general-topology/topological-spaces.md')
-rw-r--r--pages/general-topology/topological-spaces.md12
1 files changed, 6 insertions, 6 deletions
diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md
index b0b1834..cb0c30b 100644
--- a/pages/general-topology/topological-spaces.md
+++ b/pages/general-topology/topological-spaces.md
@@ -75,7 +75,8 @@ is the smallest topology on $X$ containing $\mathcal{A}$.
{% definition Basis for a Topology %}
A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$
-such that for every point $x \in X$
+such that for every point $x \in X$
+
- there exists $B \in \mathcal{B}$ such that $x \in B$,
- if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$,
then there exists a $B_3 \in \mathcal{B}$
@@ -85,6 +86,7 @@ such that for every point $x \in X$
{% theorem Topology Generated by a Basis %}
If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$,
then the topology generated by $\mathcal{B}$ equals
+
- the collection of all subsets $S \subset X$ with the property
that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$
such that $x \in B$ and $B \subset S$;
@@ -125,7 +127,7 @@ then the topology generated by $\mathcal{S}$ equals
Suppose $(X,\mathcal{T})$ is a topological space.
A subset $S$ of $X$
is called *open* with respect to $\mathcal{T}$
-when it belongs to $\mathcal{T}$
+when it belongs to $\mathcal{T}$,
and it is called *closed* with respect to $\mathcal{T}$
when its complement $X \setminus S$ belongs to $\mathcal{T}$.
{% enddefinition %}
@@ -137,10 +139,8 @@ if and only if its complement is closed.
Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then
{: .mb-0 }
- $X$ and $\varnothing$ belong to $\mathcal{C}$,
-- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$,
-- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$.
+- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$,
+- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$.
{% endproposition %}
## The Subspace Topology
-
-