Let $X$ be a topological space , then is it true that every connected component of $X$ is a union of path-connected components ? I only know that for every point , its path connected component is contained in its connected component . Please help . Thanks in advance

Forgive me if this question has been asked before, but I did a quick search and nothing came up. My book (Geometry and Topology by Bredon) Defines components of a topological space $X$ as The collection of equivalence classes of the equivalence relation "$p$ and $q$ belong to a connected sub...

Question I'm struggling with the following: Let $C$ be a connected component of a compact Hausdorff space $X$ and let $U$ be an open set containing $C$. Prove that there exists a clopen set $V$ such that $C\subset V\subset U$. I think I have a solution, but what I've "proved" is stronger: ...

The tag connected-component has been created recently. (It is still listed among new tags.) At the moment there are only three questions having this tag. I do not think that a separate tag is needed. The tag connectedness explicitly mentions in the tag-info that it should serve also for question...

One context in which a distinction between $+\infty$ and $-\infty$ is important is in things like $$ \lim_{t\,\to\,+\infty} \frac 1 {1+e^t} = 0, \qquad \lim_{t\,\to\,-\infty} \frac 1 {1+e^t} = 1. $$ However, with rational functions $f(x)$ one can write \begin{align} & \lim_{x\,\to\,\infty} f(x) ...

I've noticed that one question was recently tagged filter. (It is the only question having this tag at the moment.) I was thinking about a few related tags, which might perhaps be useful; but I wanted to ask about the opinion of other users before creating any of these tags. (I might be biased, ...