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Path-Connected Components of Topological Manifolds are Open

Last updated Nov 1, 2022

# Statement

Let XX be a Topological Manifold

Definition Suppose MM is a . We say MM is a with nn if MM is a MM is a...

11/7/2022

of Manifold Dimension nn. Then every Path-Connected Component

Path-Connected Component

Definition Let XX be a . Then P⊂XP \subset X is a of XX if PP is a (wrt...

11/7/2022

C⊂XC \subset X is Open

Open

Definition Suppose (X,τ)(X, \tau) is a . Then U⊂XU \subset X is if U∈τU \in \tau....

11/7/2022

.

# Proof

Topological Manifolds are Locally Path-Connected

Topological Manifolds are Locally Path-Connected

Statement Let MM be a of nn. MM is . Proof Consider a B⊂MB \subset M. Because ...

11/7/2022

and Path-Connected Components of Locally Path-Connected Spaces are Open

Path-Connected Components of Locally Path-Connected Spaces are Open

Statement Let XX be a that is . Then every P⊂XP \subset X is . Proof Let P⊂XP \subset X be...

11/7/2022

. ■\blacksquare

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Path-Connected Components of Topological Manifolds are Open/Topological Manifold/Manifold DimensionPath-Connected ComponentOpenTopological Manifolds are Locally Path-ConnectedPath-Connected Components of Locally Path-Connected Spaces are Open