# Questions tagged [locally-connected]

For questions on locally connected topological spaces. A topological space is called locally connected if every neighborhood of every point contains a connected open neighborhood.

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### Number of components of an open subspace of a compact locally connected space

In a locally connected space $X$, one can show that the connected components of any open subspace $U\subseteq X$ are all open in $X$ (cf. Theorem 25.3 in Munkres' Topology 2e). Therefore, if $X$ is ...
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### Is an open connected subspace of $\mathbb{R^2}$ locally path-connected?

I want to proof the following statement : Let X be an open connected subspace of $\mathbb{R^2}$. Show that X is also path connected. The standard way to prove this problem from what I at least saw was ...
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### Is arbitrary intersection of locally connected topologies locally connected?

Let $X$ be a set and $\{\mathcal{T}_i \subset \mathcal{P}(X) : i \in I\}$ be an arbitrary set of locally connected topologies in $X$; it may or may not contain all of them and $I$ may be finite or ...
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### Is the local connectedness heritable over the connected subspaces?

Definition A topological space $X$ is locally connected if each point $x\in X$ has a base of connected neighborhoods. So let be $X$ a locally connected space and let be $Y$ a connected subspace. So if ...
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### Boundaries in Spaces where Quasicomponents and Components Coincide

Let's call a space $X$ geometric if its components and quasi-components coincide. Let's also define a property called the boundary bumping property: $X$ has the boundary bumping property ("bbp&...
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### X has finitely many connected components, can we get X locally connected?

say X has finitely many connected components, can we get X locally connected? I'm thinking to prove this and then I can have every component is open. Thanks!
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### Proof Verification : Equivalent Condition for Locally Connected Space

A topological space $X$ is locally connected if for every $x$ in $X$ and for every open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$. I think it is equivalent ...
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### Local connectedness of topologist's comb

I'm using Croom's textbook, and came with the following definition of local connectedness: A space $X$ is locally connected at a point $p$ in $X$ if every open set containing $p$ contains a connected ...
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### Metric space locally compact but not uniformly locally connected

Definition: A set $M$, also in a metric space, is said to be uniformly locally connected if and only if for every $\varepsilon > 0$ there exists $\delta>0$ such that any pair of points $x, y$ of ...
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