Content deleted Content added
m →Properties: add one link |
m shortened Stack Exchange URL |
||
Line 31:
A space <math>X</math> is called '''connected im kleinen at <math>x</math>'''<ref>Willard, Definition 27.14, p. 201</ref><ref name="BBS"/> or '''weakly locally connected at <math>x</math>'''<ref>Munkres, exercise 6, p. 162</ref> if every neighborhood of <math>x</math> contains a connected neighborhood of <math>x</math>, that is, if the point <math>x</math> has a neighborhood base consisting of connected sets. A space is called '''weakly locally connected''' if it is weakly locally connected at each of its points; as indicated below, this concept is in fact the same as being locally connected.
A space that is locally connected at <math>x</math> is connected im kleinen at <math>x.</math> The converse does not hold, as shown for example by a certain infinite union of decreasing [[broom space]]s, that is connected im kleinen at a particular point, but not locally connected at that point.<ref name="SS-119.4">Steen & Seebach, example 119.4, p. 139</ref><ref name="Munkres-ex7-p162">Munkres, exercise 7, p. 162</ref><ref>{{cite web |title=Show that X is not locally connected at p |url=https://math.stackexchange.com/
A space <math>X</math> is said to be '''path connected im kleinen at <math>x</math>'''<ref name="BBS">{{cite journal |last1=Björn |first1=Anders |last2=Björn |first2=Jana |last3=Shanmugalingam |first3=Nageswari |title=The Mazurkiewicz distance and sets that are finitely connected at the boundary |journal=Journal of Geometric Analysis |volume=26 |year=2016 |issue=2 |pages=873–897 |doi=10.1007/s12220-015-9575-9 |arxiv=1311.5122|s2cid=255549682 }}, section 2</ref> if every neighborhood of <math>x</math> contains a path connected neighborhood of <math>x</math>, that is, if the point <math>x</math> has a neighborhood base consisting of path connected sets.
A space that is locally path connected at <math>x</math> is path connected im kleinen at <math>x.</math> The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above. However, if a space is path connected im kleinen at each of its points, it is locally path connected.<ref>{{cite web |title=Definition of locally pathwise connected |url=https://math.stackexchange.com/
==First examples==
| |||