Thank you Doodledog.
Sexuality has nothing to do with this either, or at least not on the straight or LGB spectrum. Some transwomen identify as lesbians, as they are sexually attracted to women (ie they are heterosexual males). Most, if not all of these are AGF (autogynephiles) who are sexually excited at the thought of being women, and they can get aggressive if women refuse to have sex with them, as refusal to believe that they are female 'bursts their bubble'.
Quite apart from the fact that women's spaces are being lost, with the attendant dangers we have seen in the recent hospital rape, men insisting that they are women means that they are more likely to win at sport, and if they are counted as female in statistics even fewer things are likely to accommodate women, or to pay attention to their needs.
I remember when one of the first women's refuges was set up in Richmond, Surrey. A virtually derelict riverside building was taken over as a temporary shelter. I lived opposite it. My then partner who was handy with a screwdriver and familiar with electrical wiring, etc, volunteered to set up some temporary heating and lighting (he was one of those men who supported what we called, at the time, 'Women's Liberation'). The building was a mess, and I remember distinctly him saying words to the effect that it was ridiculous that there were no established, legal, safe places for abused women and children. Well now there are, and they were hard-won, and not without controversy and opposition either. Yet, here we are having to fight the fight again.
A trans woman with a penis may well not be a threat to women and girls, but that's not the point which is that the abused women and girls need a space that is run by and for them. Trans women also suffer abuse and violence and they need safe spaces too - but why does it have to be women's space? Why is it considered transphobic to recognise that a trans woman's experiences and a woman's experiences will not be the same, and need different approaches in counselling and care?
When we started on this journey of 'Liberation' from male dominance, I never expected that one day we'd have have to defend not only our rights, but the very concept of being a woman.
There was controversy over the occupation in Richmond, some locals were outraged. But there was also support - practical support by way of donations of clothing, bedding, toys and money. Ultimately the women were turfed out (it was an illegal occupation) and a place was found in Chiswick... and that's now part of history. The riverside was redeveloped and there may not be many people left in Richmond who remember this episode. I no longer live there.