topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
It is not generally true that a topological space is the disjoint union space (coproduct in Top) of its connected components. The spaces such that this is true for all open subspaces are the locally connected topological spaces.
(locally connected topological space)
A topological space is locally connected if every point has a neighborhood basis of connected open subsets.
(alternative characterizations of local connectivity)
For a topological space, then the following are equivalent:
every connected component of every open subspace of is open;
every open subset, as a topological subspace, is the disjoint union space (coproduct in Top) of its connected components.
In particular, in a locally connected space, every connected component is a clopen subset; hence connected components and quasi-components coincide.
1) 2)
Assume is locally connected, and let be an open subset with a connected component. We need to show that is open.
Consider any point . Since then also , the defintion of local connectedness, def. , implies that there is a connected open neighbourhood of . Observe that this must be contained in , for if it were not then were a larger open connected open neighbourhood, contradicting the maximality of the connected component .
Hence is a union of open subsets, and hence itself open.
2) 3)
Now assume that every connected component of every open subset is open. Since the connected components generally consitute a cover of by disjoint subsets this means that now they for an open cover by disjoint subsets. But by forming intersections with the cover this implies that every open subset of is the disjoint union of open subsets of the connected components (and of course every union of open subsets of the connected components is still open in ), which is the definition of the topology on the disjoint union space of the connected components.
3) 1)
Finally assume that every open subspace is the disjoint union of its connected components. Let be a point and a neighbourhood. We need to show that contains a connected neighbourhood of .
But, by definition, contains an open neighbourhood of and by assumption this decomposes as the disjoint union of its connected components. One of these contains . Since in a disjoint union space all summands are open, this is the required connected open neighbourhod.
Every discrete topological space is locally connected.
(Euclidean space is locally connected)
For the Euclidean space (with its metric topology) is locally connected.
By nature of the Euclidean metric topology, every neighbourhood of a point contains an open ball containing . Moreover, every open ball clearly contains an open cube, hence a product space of open intervals which is still a neighbourhood of .
Now intervals are connected (by this example) and products of connected spaces are connected (by this example). This shows that ever open neighbourhood contains a connected neighbourhood.
(open subspace of locally connected space is locally connected)
Every open subspace of a locally connected space is itself locally connected
The topologist's sine curve is connected but not locally connected.
Examples of locally connected spaces include topological manifolds.
Finally,
A space is totally disconnected topological space if its connected components are precisely the singletons of .
In other words, a space is totally disconnected if its coreflection into is discrete. Such spaces recur in the study of Stone spaces.
The category of totally disconnected spaces is a reflective subcategory of . The reflector sends a space to the space whose points are the connected components of , endowed with the quotient topology induced by the projection . Details may be found at totally disconnected space.
Let be the full subcategory inclusion of locally connected spaces into all of Top. The following result is straightforward but useful.
is a coreflective subcategory of , i.e., the inclusion has a right adjoint . For a given space, has the same underlying set as and the coarsest locally connected topology that is finer than the original topology on .
Being a coreflective category of a complete and cocomplete category, the category is also complete and cocomplete. Of course, limits and particularly infinite products in are not calculated as they are in ; rather one takes the limit in and then retopologizes it according to Theorem . (For finite products of locally connected spaces, we can just take the product in – the result will be again locally connected.)
Let be the underlying set functor, and let be the functors which assign to a set the same set equipped with the codiscrete and discrete topologies, respectively. Let be the functor which assigns to a locally connected space the set of its connected components.
There is an adjoint quadruple of adjoint functors
and moreover, the functor preserves finite products.
While is not a topos, this is the adjunction structure as on a cohesive topos.
The proof is largely straightforward; we point out that the continuity of the unit is immediate from a locally connected space’s being the coproduct of its connected components. As for preserving finite products, write locally connected spaces , as coproducts of connected spaces
then their product in coincides with their product in , and is
where each summand is connected by Result . From this it is immediate that preserves finite products.
Accordingly the category of sheaves on a locally connected space is a locally connected topos. For related discussions, see also cohesive topos.
A quotient space of a locally connected space is also locally connected.
Suppose is a quotient map, and let be an open neighborhood of . Let be the connected component of in ; we must show is open in . For that it suffices that be open in , or that each is an interior point. Since is locally connected, the connected component of in is open, and the subset is connected, and therefore (as is the maximal connected subset of containing ). Hence , proving that is interior to , as desired.
The conclusion does not follow if is merely surjective; e.g., there is a surjective (continuous) map from to (a version of) the Warsaw circle, but the latter is not locally connected.
Last revised on January 29, 2021 at 06:59:43. See the history of this page for a list of all contributions to it.