Topological spaces whose quasi-components are connected Let ${\mathcal X}$ be the category of topological spaces whose quasi-components are connected (and all continuous functions between them).
I know that compact Hausdorff spaces are in ${\mathcal X}$. I also know that locally connected spaces are in ${\mathcal X}$. What else is known about ${\mathcal X}$? (I could not find any additional information in the web.) For example: Is it complete? Is it (co)reflective as a subcategory of topological spaces? ...
Thanks.
 A: Some thoughts:

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*A carefull analysis of the proof of the fact that Hausdorff compact spaces are in $\mathcal{X}$ shows that it is true for normal (not necessarily Hausdorff) compact spaces.


*Every space with clopen quasi-components (equivalently a topological sum of connected spaces) is in $\mathcal{X}$. This includes locally connected spaces as well as connected spaces. It also means that we can take any connected spaces with no other restrictions, take their sum, and the result will be in our class.


*A space $X$ is in our class if and only if for every (necessarily closed) component $C$ and every $x ∈ X ∖ C$ there is a clopen set separating them. This way we can view the property as a kind of separation axiom. Note that $x$ is contained in its closed component, which is disjoint from $C$, so we may equvialently talk about clopen separation of disjoint (different) components. Also note that every ultranormal, zero-dimensional, or totally separated space is in our class (these are defined by clopen separation of disjoint closed sets / points and closed sets / points by clopen sets).


*For every $X$ we may form the hereditary disconnected reflection $X \to X/{\sim}$ by collapsing the components to points, and the totally separated reflection $X \to X/{≈}$ by collapsing the quasi-components. $X ∈ \mathcal{X}$ if and only if the reflections coincide, or equivalently if the hereditarily disconnected space $X/{\sim}$ is totally separated. So the property only depends on $X/{\sim}$.


*Regarding reflexivity, let $X'$ be the “totally separated modulo $T_0$” reflection, i.e. we take the totally separated reflection and pull the topology back to the underlying set of $X$, so the $T_0$ reflection $X'$ is the totally separated reflection of $X$. We have that $X'$ is in $\mathcal{X}$, if the reflection exists, it is a topology between $X$ and $X'$.
Similarly, let $X''$ be finer topology on $X$ making all components clopen. If the coreflection in $\mathcal{X}$ exists, it is a topoloy between $X''$ and $X$.
I would guess that $\mathcal{X}$ is not reflexive and not coreflexive, but it is just a guess.
